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UNDERSTANDING FINANCIAL CRISES

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UnderstandingFinancial Crises

FRANKLIN ALLEN

and

DOUGLAS GALE

1

3Great Clarendon Street, Oxford OX2 6DP

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Preface

This book has grown out of a series of papers written over a number ofyears. Our paper “Liquidity Preference, Market Participation, and Asset PriceVolatility” was actually begun by one of us in 1988, although it appeared in1994. Our interest in bank runs and financial crises began with “OptimalFinancial Crises,” and this led to further studies on the welfare economics ofcrises. Each paper seemed to leave questions unanswered and led to new paperswhich led to new questions.

When one of us was invited to give the Clarendon Lectures in Finance, itseemed the right time to begin the task of synthesizing the ideas presentedin a variety of different places using different models. Our aim was to makethese ideas accessible to a wider audience, including undergraduates and policymakers in central banks and international organizations, and also to put themin a coherent framework that might make them more useful for graduatestudents and researchers. This is far from being the last word on the subject,but it may provide a set of tools that will be helpful in pursuing the many openquestions that remain.

Over the years we have hadmany opportunities to present our work in semi-nars and at conferences andhave benefited from the comments and suggestionsof many economists. In particular, we would like to thank Viral Acharya,Christina Bannier, Michael Bordo, Patrick Bolton, Mike Burkart, Mark Carey,Elena Carletti, Michael Chui, Marco Cipriani, Peter Englund, Prasanna Gai,Gary Gorton, Antonio Guarino, Martin Hellwig, Marcello Pericoli, Glen Hog-garth, Jamie McAndrews, Robert Nobay, Önür Ozgur, João Santos, MassimoSbracia, Hyun Song Shin, Giancarlo Spagnolo, Xavier Vives, David Webb,Andrew Winton, and Tanju Yorulmazer.

We have included some of these topics in graduate courses we taught at NewYork University and Princeton University. Once we began writing the book,we were fortunate to have the opportunity to present lecture series at the Bankof England, the Banca d’Italia, the Stockholm School of Economics, and theUniversity of Frankfurt. We developed an undergraduate course on financialcrises at NYU based on the manuscript of the book. We are very gratefulto the undergraduates who allowed us to experiment on them and rose to thechallenge presented by thematerial. Antonio Guarino used several chapters for

vi Preface

an undergraduate course at University College London and offered us manycomments and corrections.

We are sure there are others whom we may have forgotten, but whosecontributions and encouragement helped us greatly. Thanks to all of you.

Contents

1. History and institutions 1

1.1 Introduction 11.2 Historical crises in Europe and the US 21.3 Crises and stock market crashes 51.4 Currency and twin crises 91.5 Crises in different eras 101.6 Some recent crises 14

1.6.1 The Scandinavian crises 141.6.2 Japan 151.6.3 The Asian crisis 151.6.4 The Russian crisis and long term capital

management (LTCM) 161.6.5 The Argentina crisis of 2001–2002 17

1.7 The costs of crises 181.8 Theories of crises 191.9 Concluding remarks 23

2. Time, uncertainty, and liquidity 27

2.1 Efficient allocation over time 272.1.1 Consumption and saving 272.1.2 Production 36

2.2 Uncertainty 402.2.1 Contingent commodities and risk sharing 402.2.2 Attitudes toward risk 442.2.3 Insurance and risk pooling 482.2.4 Portfolio choice 49

2.3 Liquidity 522.4 Concluding remarks 57

3. Intermediation and crises 58

3.1 The liquidity problem 593.2 Market equilibrium 603.3 The efficient solution 643.4 The banking solution 723.5 Bank runs 74

viii Contents

3.6 Equilibrium bank runs 763.7 The business cycle view of bank runs 823.8 The global games approach to finding a unique

equilibrium 903.9 Literature review 943.10 Concluding remarks 96

4. Asset markets 99

4.1 Market participation 994.2 The model 1034.3 Equilibrium 104

4.3.1 Market-clearing at date 1 1074.3.2 Portfolio choice 109

4.4 Cash-in-the-market pricing 1104.5 Limited participation 114

4.5.1 The model 1164.5.2 Equilibrium 1174.5.3 Equilibrium with full participation 1204.5.4 Full participation and asset-price volatility 1204.5.5 Limited participation and asset-price volatility 1214.5.6 Multiple Pareto-ranked equilibria 123

4.6 Summary 124

5. Financial fragility 126

5.1 Markets, banks, and consumers 1285.2 Types of equilibrium 132

5.2.1 Fundamental equilibrium with no aggregateuncertainty 133

5.2.2 Aggregate uncertainty 1355.2.3 Sunspot equilibria 1405.2.4 Idiosyncratic liquidity shocks for banks 1425.2.5 Equilibrium without bankruptcy 1445.2.6 Complete versus incomplete markets 146

5.3 Relation to the literature 1475.4 Discussion 148

6. Intermediation and markets 153

6.1 Complete markets 1556.2 Intermediation and markets 164

6.2.1 Efficient risk sharing 1656.2.2 Equilibrium with complete financial markets 167

Contents ix

6.2.3 An alternative formulation of complete markets 1706.2.4 The general case 1726.2.5 Implementing the first best without complete

markets 1776.3 Incomplete contracts 181

6.3.1 Complete markets and aggregate risk 1826.3.2 The intermediary’s problem with incomplete

markets 1866.4 Conclusion 188

7. Optimal regulation 190

7.1 Capital regulation 1917.1.1 Optimal capital structure 1947.1.2 Models with aggregate uncertainty 199

7.2 Capital structure with complete markets 2017.3 Regulating liquidity 204

7.3.1 Comparative statics 2067.3.2 Too much or too little liquidity? 209

7.4 Literature review 2137.5 Concluding remarks 213

8. Money and prices 216

8.1 An example 2188.2 Optimal currency crises 2238.3 Dollarization and incentives 2268.4 Literature review 2288.5 Concluding remarks 232

9. Bubbles and crises 235

9.1 Agency problems and positive bubbles 2379.1.1 The risk-shifting problem 2389.1.2 Credit and interest rate determination 2439.1.3 Financial risk 2459.1.4 Financial fragility 246

9.2 Banking crises and negative bubbles 2479.2.1 The model 2479.2.2 Optimal risk sharing 2489.2.3 Optimal deposit contracts 2519.2.4 An asset market 2529.2.5 Optimal monetary policy 256

9.3 Concluding remarks 258

x Contents

10. Contagion 260

10.1 Liquidity preference 26310.2 Optimal risk sharing 26610.3 Decentralization 26810.4 Contagion 274

10.4.1 The liquidation “pecking order” 27510.4.2 Liquidation values 27610.4.3 Buffers and bank runs 27710.4.4 Many regions 280

10.5 Robustness 28010.6 Containment 28110.7 Discussion 28210.8 Applications 284

10.8.1 Upper and Worms (2004) 28410.8.2 Degryse and Nguyen (2004) 29010.8.3 Cifuentes, Ferrucci, and Shin (2005) 291

10.9 Literature review 29310.10 Concluding remarks 295

Index 299

1

History and institutions

1.1 INTRODUCTION

What happened in Asia in 1997? Countries such as South Korea, Thailand,Indonesia, Singapore, and Hong Kong whose economies had previously beenthe envy of the world experienced crises. Banks and other financial intermedi-aries were put under great strain and in many cases collapsed. Stock marketsand currencies plunged. Their real economies were severely affected and theirGDPs fell significantly. What were the causes of these dramatic events?

To many people these crises were a new phenomenon. There had been crisesin other countries such as Mexico and Brazil but these could be attributed toinconsistent government macroeconomic policies. In those cases taxes weretoo small relative to government expenditures to maintain a fixed exchangerate. This was not the case for the Asian crisis. Other causes were lookedfor and found. The institutions in these countries were quite different fromthose in the US. Many had bank-based financial systems. There was littletransparency either for banks or corporations. Corporate governance operatedin a quite different way. In many cases it did not seem that managers’ interestswere aligned with those of shareholders. In some countries such as Indonesiacorruption was rife. These factors were seen by many as the cause of the crises.However, they had all been present during the time that these countries wereso successful.

Others blamed guarantees to banks and firms by governments or implicitpromises of “bail-outs” by organizations such as the International Mone-tary Fund (IMF). Rather than inconsistent macroeconomic policies being theproblem, bad microeconomic policies were the problem. Either way it wasgovernments and international organizations that were to blame.

In this book we will argue that it is important not to take too narrow a viewof crises. They are nothing new. They have not been restricted to emergingeconomies even in recent times. The Scandinavian crises of the early 1990’sare examples of this. Despite having sophisticated economies and institutions,Norway, Sweden and Finland all had severe crises. These were similar in manyways to what happened in theAsian crisis of 1997. Banks collapsed, asset prices

2 Chapter 1. History and institutions

plunged, currencies came under attack and their value fell. Output was severelyaffected.

Taking an historical view the period from 1945–1971 was exceptional. Therewere no banking crises anywhere in the world, apart fromone in Brazil in 1962.There were currency crises when exchange rates were pegged at the wronglevels but that was all. Going back to the first half of the twentieth century andbefore there were many examples of financial crises. The stock market crashof 1929, the banking crises of the early 1930’s and the Great Depression weresome of the most dramatic episodes. There were many others, particularly inthe US in the last half of the nineteenth century when it had no central bank.In Europe crises were much less frequent. The Bank of England had learnedto prevent crises and the last one there was the Overend & Gurney crisis of1866. Other central banks also learned to prevent crises and their incidencewas significantly reduced. Prior to that crises were endemic in Europe as well.

Particularly after the experience of the Great Depression in the period priorto 1945–1971, crises were perceived as a market failure. It was widely agreedthey must be avoided at all costs. The reform of the Federal Reserve System inthe early 1930’s and the extensive regulation of the financial system that wasput in place in the US were part of this mindset. In other countries financialregulation went even farther. Governments controlled the allocation of fundsto different industries through state-owned banks or heavily regulated banks.This extensive regulation was the cause of the virtual disappearance of bankingcrises from 1945–1971.

However, the elimination of crises came at a cost. Because of the extensiveregulation and government intervention the financial systemceased to performits basic function of allocating investment. There were many inefficiencies as aresult. This led to calls for deregulation and the return of market forces to theallocation of investment. As a result crises have returned. Bordo et al. (2000)find that the frequency of crises in the recent period since 1971 is not thatdifferent from what it was before 1914.

We start in this chapterwith an historical review of crises and the institutionsinvolved. This provides a background for the theories that are subsequentlydeveloped.

1.2 HISTORICAL CRISES IN EUROPE AND THE US

Prior to the twentieth century banking panics occurred frequently. Kindle-berger (1993, p. 264) in his book recounting the financial history of WesternEurope points out that financial crises have occurred at roughly 10 year

1.2 Historical Crises in Europe and the US 3

intervals over the last 400 years. Panics were generally regarded as a bad thingbecause they were often associated with significant declines in economic activ-ity. Over time one of the main roles of central banks has become to eliminatepanics and ensure financial stability. It has been a long and involved process.The first central bank, the Bank of Sweden, was established over 300 years agoin 1668. The Bank of England was established soon after. It played an espe-cially important role in the development of effective stabilization policies inthe eighteenth and nineteenth centuries. The last true panic in the UK was theOverend & Gurney crisis of 1866.

In his influential book Lombard Street, Bagehot (1873) laid out his famousprinciples of how a central bank should lend to banks during a crisis.

• Lend freely at a high rate of interest relative to the pre-crisis period but onlyto borrowers with good collateral (i.e. any assets normally accepted by thecentral bank).

• The assets should be valued at between panic and pre-panic prices.• Institutions without good collateral should be allowed to fail.

Bordo (1986) documents that for the period 1870–1933 there were very fewbanking panics in the UK, Germany, and France. Kindleberger (1993) pointsout that many British economists ascribe the absence of crises in the UK tocentral banking experience gained by the Bank of England and their abilityto skillfully manipulate discount rates. However, France also experienced nofinancial crises from 1882–1924 despite leaving its discount rate constant formany decades. Kindleberger suggests that France was perhaps stabilized byEngland.

The US took a different tack. Alexander Hamilton was influenced by Britishexperience with the Bank of England and after the revolution advocated alarge federally chartered bank with branches all over the country. This led tothe foundation of the First Bank of the United States (1791–1811) and laterthe Second Bank of the United States (1816–1836). However, there was con-siderable distrust of the concentration of power these institutions represented.In a report on the Second Bank, John Quincy Adams wrote “Power for good, ispower for evil, even in the hands of Omnipotence” (Timberlake 1978, p. 39).The controversy came to a head in the debate on the re-chartering of theSecond Bank in 1832. Although the bill was passed by Congress it was vetoedby President Jackson and the veto was not overturned. Since then there hasbeen a strong bias toward decentralization of the banking system and an aver-sion to powerful institutions of any kind. There was no central bank in the USfrom 1836 until 1914.


Throughout the nineteenth century the US banking system was highly frag-mented and unlike every other industrializing country the US failed to developnationwide banks with extensive branch networks. Prior to the CivilWar, stateswere free to regulate their own banking systems and there was no national sys-tem. Many states adopted a “free banking” system which allowed free entry.There were serious banking panics in 1837 and 1857 and both were followedby depressions and significant economic disruption.

The advent of the Civil War in 1861 and the need to finance it sig-nificantly changed the role of the Federal Government in the financialsystem. The National Bank Acts of 1863 and 1864 set up a national bank-ing system. They granted limited powers to banks. In particular, the 1864Act was interpreted as confining each to a single location. When thequestion of whether banks could hold equity arose, the Supreme Courtruled that since the 1864 Act had not specifically granted this right theycould not.

The creation of the National Banking system did not prevent the problemof panics and the associated economic disruption and depressions. There werepanics in 1873, 1884, 1893 and 1907. Table 1.1, which is from Gorton (1988),shows the banking crises that occurred repeatedly in theUSduring theNationalBanking Era from 1863–1914. The first column shows the business cyclesidentified by the National Bureau of Economic Research (NBER). The firstdate is the peak of the cycle and the second is the trough. The second columnshows the date on which panics occurred. In a banking panic people worry

Table 1.1. National Banking Era panics.

NBER cyclePeak–Trough

Panic date %�(Currency/deposit)∗

%� Pigiron†

Oct. 1873–Mar. 1879 Sep. 1873 14.53 −51.0Mar. 1882–May 1885 Jun. 1884 8.80 −14.0Mar. 1887–Apr. 1888 No Panic 3.00 −9.0Jul. 1890–May 1891 Nov. 1890 9.00 −34.0Jan. 1893–Jun. 1894 May 1893 16.00 −29.0Dec. 1895–Jun. 1897 Oct. 1896 14.30 −4.0Jun. 1899–Dec. 1900 No Panic 2.78 −6.7Sep. 1902–Aug. 1904 No Panic −4.13 −8.7May 1907–Jun. 1908 Oct. 1907 11.45 −46.5Jan. 1910–Jan. 1912 No Panic −2.64 −21.7Jan. 1913–Dec. 1914 Aug. 1914 10.39 −47.1

∗Percentage change of ratio at panic date to previous year’s average.†Measured from peak to trough.(Adapted from Table 1, Gorton 1988, p. 233)

1.3 Crises and Stock Market Crashes 5

about the soundness of the banks they have deposited their funds in. As aresult they withdraw their money and hold it in the form of currency. Thethird column shows the percentage change in the ratio of currency to deposits.It is a measure of the severity of a banking panic. The higher the change inthe currency/deposit ratio, the more serious is the crisis. It can be seen thatthe panics of 1873, 1893, 1896, and 1907 were particularly severe. The finalcolumn shows how much the production of pig iron changed from the peakof the cycle to the trough. GDP figures for this period have not been reliablycompiled. Economic historians often use production of pig iron as a proxyfor GDP. The final column is therefore meant to indicate how serious therecessions were. It can be seen that the troughs occurring after the panics of1873, 1890, 1893, 1907, and 1914 were particularly severe.

After the crisis of 1907, a European banker summedupEuropean frustrationwith the inefficiencies of the U.S. banking system by declaring the US was “agreat financial nuisance” (Studenski and Krooss 1963, p. 254). The severity ofthe recession following the 1907 banking panic led to a debate on whether ornot a central bank should be established in the US. The National MonetaryCommission investigated this issue and finally in 1914 the Federal ReserveSystem was established.

The initial organization of the Federal Reserve System differed from that ofa traditional central bank like the Bank of England. It had a regional structureand decision making power was decentralized. During the years after its cre-ation it did not develop the ability to prevent banking panics. In 1933 therewas another major banking panic which led to the closing of banks for anextended period just after President Roosevelt took office. The problems facedby the banking system led to the Glass–Steagall Act of 1933, which introduceddeposit insurance and required the separation of commercial and investmentbanking operations. The Banking Act of 1935 extended the powers of the Fed-eral Reserve System and changed the way it operated. These reforms finallyeliminated the occurrence of banking panics almost seventy years after thishad happened in the UK.

1.3 CRISES AND STOCK MARKET CRASHES

So farwehave focusedonbanking crises.Oftenbanking crises and stockmarketcrashes are closely intertwined. For example,Wilson et al. (1990) consider fourmajor banking panics accompanied by stock market crashes in the US duringthe National Banking Era. These are the crises of September 1873, June 1884,July 1893, and October 1907.


Why was there a link between banking panics and stock market crashes? Asmentioned above banks were not able to hold equity so it might be thoughtthat movements in the stock market would be independent of banks’ pol-icies. In fact this was not the case. To see why, it is necessary to have someunderstanding of the link between banks and the stock market during thisperiod.

Banks must hold liquid reserves in case customers wish to withdraw cashfrom their accounts. All banks hold some reserves in the form of currency.In addition a large proportion of reserves were held in the form of interbankbalances. In practice, most banks had deposits in New York City banks. Thereason banks held interbank deposits rather than cash was that they paidinterest. The NewYork City banks could pay attractive rates of interest becausethey lent a large proportion of these funds in the call loan market at the stockexchange in New York. The loans were used to buy stocks on margin (i.e. thestocks were bought with borrowed money). They were referred to as call loansbecause they were payable on demand. The borrowers could either obtainfunds to repay their call loans by taking out other loans or if necessary theycould sell the securities the original call loans were used to purchase. These callloans constituted a large part of New York banks’ assets. For example, Sprague(1910,p. 83) reports that on September 12, 1873, 31 percent of NewYork banks’loans were call loans.

Agriculture was much more important during the National Banking Erathan it is today. During the Spring planting and Autumn harvesting banks infarming areas required cash. Because of the random nature of these demandsfor cash it was difficult for the New York City banks to plan with certaintywhat their liquidity needs would be. When liquidity needs were high the NewYork City banks would demand repayment of their call loans. The borrowersmight be forced to sell the securities they had purchased on margin. A waveof selling could cause prices to fall if those participating in the market on thebuy side had limited amounts of cash. In other words there could be a crash inprices.

Wilson, Sylla, and Jones investigate stock returns and their volatility dur-ing the panic and crash periods of 1873, 1884, 1893, and 1907. Table 1.2shows the 25 lowest and 25 highest stock monthly price changes between1866 and 1913. Four of the eight lowest returns occurring during this periodwere during panic months. Apart from May 1880, which is not associatedwith a banking panic, all the others from the nine lowest returns are aroundpanics. Notice also from the highest stock returns that there is some ten-dency for stocks to rally two or three months after a crisis. December 1873

1.3 Crises and Stock Market Crashes 7

Table 1.2. The 25 lowest and 25 highest stock price changes1866–1913 (from Table 1 of Wilson et al. 1990).

Year Month Lowest Rank Year Month Highestreturn return

1907 10 −10.8514% 1 1879 10 10.8824%1907 3 −9.7987 2 1901 6 9.96781893 7 −9.4340 3 1873 12 9.53851893 5 −8.8993 4 1901 4 8.44371873 10 −8.6721 5 1891 9 8.06051884 5 −8.5575 6 1900 11 7.85121880 5 −7.9137 7 1899 1 7.69231873 9 −7.7500 8 1906 8 7.40741907 8 −7.4809 9 1877 8 6.98691890 11 −7.3350 10 1898 5 6.81201877 6 −7.1730 11 1893 9 6.68691877 4 −7.0588 12 1897 8 6.68521899 12 −6.7308 13 1896 11 6.66671901 7 −6.7251 14 1908 11 6.60661896 7 −6.6092 15 1884 8 6.40671869 9 −6.4913 16 1885 11 6.31311884 6 −6.4171 17 1898 12 6.30841876 9 −6.0127 18 1877 9 6.12241877 2 −5.9441 19 1881 1 5.95741907 11 −5.8052 20 1904 10 5.94231895 12 −5.6911 21 1900 12 5.93871903 6 −5.5556 22 1885 10 5.88241896 8 −5.5385 23 1895 5 5.69801911 9 −5.4201 24 1882 7 5.68931877 3 −5.2045 25 1885 8 5.5710

is the third highest return, September 1993 is the eleventh highest, andAugust 1884 is the fifteenth highest. It is not just stocks where this effectis found. Bonds and commercial paper show similar patterns of returns.Returns are low during the panic and then rebound in the months after thepanic.

Table 1.3 shows the top 50 months of volatility for stocks between 1866 and1913. These volatilities are calculated by including the annualized standarddeviation of returns using the current month and nine of the previous 11months with the two discarded being the ones with the highest and lowestreturns. The greatest volatility seems to occur in the year following the panicwith peak stock price volatility coming 2–7 months after the panic.


Table 1.3. The top 50 months of volatility for stocks1866–1913 (from Table 5 of Wilson et al. 1990).

Rank Stocks

Year Mo. Stocks

1 1908 5 16.24332 1908 6 15.68013 1908 7 15.62394 1908 4 15.45905 1908 2 15.05096 1908 1 15.01797 1901 7 15.00788 1878 1 14.21829 1877 10 14.1960

10 1877 12 14.1921

11 1877 11 14.184112 1873 12 14.146113 1908 3 13.972214 1901 8 13.769515 1901 10 13.764516 1877 8 13.745917 1877 9 13.723818 1907 12 13.549719 1893 9 13.527320 1908 9 13.0782

21 1908 8 13.065822 1901 9 13.051923 1878 2 13.020624 1896 11 12.815325 1894 4 12.564126 1901 5 12.421427 1894 3 12.383628 1901 11 12.354329 1891 9 12.207930 1884 8 12.1837

31 1898 5 12.043032 1901 12 12.001433 1901 6 11.952634 1902 1 11.894735 1878 3 11.841536 1893 12 11.815437 1874 8 11.812738 1902 2 11.804239 1880 5 11.788040 1898 6 11.7863

1.4 Currency and Twin Crises 9

Table 1.3. (Continued)

Rank Stocks

Year Mo. Stocks

41 1874 7 11.780242 1874 6 11.757143 1874 5 11.744244 1874 4 11.713245 1893 11 11.704046 1874 3 11.506847 1894 2 11.504048 1874 2 11.491449 1901 4 11.448050 1902 3 11.4422

1.4 CURRENCY AND TWIN CRISES

Many of the crises in the nineteenth and early twentieth century were inter-national in scope. For example, the crisis of 1873 had an extensive impact inAustria and Germany as well as in the US and in a number of emerging coun-tries such as Argentina. In fact the 1873 crisis ended a wave of lending thatoccurred in the 1850’s and 1860’s to finance railroads in Latin America (Bordoand Eichengreen 1999). These international dimensions led to a flow of fundsbetween countries and this in turn could cause a currency crisis.When bankingcrises and currency crises occur together there is said to be a twin crisis.

Prior to the FirstWorldWar countries had a strong commitment to the goldstandard. If a country suffered an outflow of funds it might leave the goldstandard but it was generally expected to resume after some time had passed.This lessened the effect of currency crises as investors believed the value of thecurrency would eventually be restored. Between the wars, commitment to thegold standardwasweakened.As a result banking and currency crises frequentlyoccurred together. These twin crises are typically associated with more severerecessions than banking or currency crises occurring on their own.

After the Second World War the Bretton Woods system of fixed exchangerateswas established. Strong banking regulations and controlswere put in placethat effectively eliminated banking crises. Currency crises continued to occur.Due to the extensive use of capital controls their nature changed. During thisperiod they were typically the result of macroeconomic and financial policiesthat were inconsistent with the prevailing exchange rate. After the collapse ofthe Bretton Woods system in the early 1970’s banking crises and twin crisesreemerged as capital controls were relaxed and capital markets became global.


1.5 CRISES IN DIFFERENT ERAS

Bordo et al. (2000, 2001) have addressed the question of how recent crisessuch as the European Monetary System crisis of 1992–1993, the Mexican crisisof 1994–1995, the Asian crisis of 1997–1998, the Brazilian crisis of 1998, theRussian crisis of 1998, and the Argentinian crisis of 2001 compare with earliercrises. They identify four periods.

1. Gold Standard Era 1880–19132. The Interwar Years 1919–19393. Bretton Woods Period 1945–19714. Recent Period 1973–1997

As we shall see there are a number of similarities between the periods butalso some important differences. They consider 21 countries for the first threeperiods and then for the recent period give data for the original 21 as well asan expanded group of 56.

The first issue is how to define a crisis. They define a banking crisis asfinancial distress that is severe enough to result in the erosion of most or allof the capital in the banking system. A currency crisis is defined as a forcedchange in parity, abandonment of a pegged exchange rate or an internationalrescue. The second issue is how to measure the duration of a crisis. To do thisthey compute the trend rate of GDP growth for five years before. The durationof the crisis is the amount of time before GDP growth returns to its trend rate.Finally, the depth of the crisis is measured by summing the output loss relativeto trend for the duration of the crisis.

Figure 1.1 shows the frequency of crises in the four periods. Comparing thedata with the original 21 countries it can be seen that the interwar years arethe worst. This is perhaps not surprising given that this was when the GreatDepression occurred. Banking crises were particularly prevalent during thisperiod relative to the other periods.

It can be seen that the BrettonWoods period is very different from the otherperiods. As mentioned above, after the Great Depression policymakers in mostcountries were so determined not to allow such an event to happen again thatthey imposed severe regulations or brought the banks under state control toprevent them from taking much risk. As a result banking crises were almostcompletely eliminated. There was one twin crisis in Brazil in 1962 but apartfrom that there were no other banking crises during the entire period. Therewere frequent currency crises but as we have seen these were mostly situationswhere macroeconomic policies were inconsistent with the level of the fixedexchange rates set in the Bretton Woods system.

1.5 Crises in Different Eras 11

14

12

10

8

6

4

2

01880–1913 1919–1939 1945–1971 1973–1997

(21 countries)1973–1997

(56 countries)

Banking crisesCurrency crisesTwin crisesAll crises

Freq

uenc

y (a

nnua

l % p

roba

bilit

y)

Figure 1.1. Crisis frequency, 1880–1997 (from Figure 1 of Bordo et al. 2001).

Interestingly the most benign period was the Gold Standard era from 1880to 1913. Here banking crises did occur but were fairly limited and currencyand twin crises were limited compared to subsequent periods. Since theglobal financial system was fairly open at this time, the implication is thatglobalization does not inevitably lead to crises.

The recent period is not as bad as the interwar period but is neverthelessfairly bad. Banking and twin crises are more frequent than in every periodexcept the interwar years and currency crises are much more frequent. This isespecially true if the sample of 56 countries is used as the basis of comparisonrather than the 21 countries used in the other periods. The countries that areadded to create the larger sample are mostly emerging countries. This suggeststhat emerging countries are more prone to crises and particularly to currencycrises.

Figure 1.2 confirms this. It breaks the sample into industrial countries andemerging markets. In recent years emerging countries have been particularlyprone to currency crises and twin crises. Theother interestingobservation fromFigure 1.2 is that during the interwar period it was the industrial countries thatwere particularly hard hit by crises. They were actually more prone to currencyand twin crises than the emerging countries.

Table 1.4 shows the average duration and depth of crises broken out by typeof crisis and for the different periods and samples. Perhaps the most strikingfeature of Table 1.4 is the short duration and mild effect of crises during the


Banking crises

Industrial countries

Emerging markets

30

25

20

15

10

5

0

30

25

20

15

10

5

0

1880–1913 1919–1939 1945–1971 1973–1997

1880–1913 1919–1939 1945–1971 1973–1997

Currency crises

Twin crises

All crises

Banking crises

Currency crises

All crises

Twin crises

Figure 1.2. Frequency of crises – distribution by market (from Figure 2 of Bordo et al.(2000)).

Bretton Woods period. The second distinctive feature is that twin crises aremuch worse than other crises in terms of the output lost. As might be expectedduring the interwar period the effect of crises was much more severe thanin the other periods. Although they did not last longer the cumulative lossin output is higher than in the other periods. During the Gold Standard Erathe duration and cumulative loss were not remarkable compared to the otherperiods. In recent years twin crises have lasted for a particularly long time andthe lost output is significant.

1.5 Crises in Different Eras 13

Table 1.4. Duration and depth of crises (from Table 1 of Bordo et al. 2001).

All countries 1880–1913 1919–1939 1945–1971 1973–1997 1973–199721 nations 56 nations

Average duration of crises in yearsCurrency crises 2.6 1.9 1.8 1.9 2.1Banking crises 2.3 2.4 a 3.1 2.6Twin crises 2.2 2.7 1.0 3.7 3.8All crises 2.4 2.4 1.8 2.6 2.5

Average crisis depth (cumulative GDP loss in %)Currency crises 8.3 14.2 5.2 3.8 5.9Banking crises 8.4 10.5 a 7.0 6.2Twin crises 14.5 15.8 1.7 15.7 18.6All crises 9.8 13.4 5.2 7.8 8.3

Notes: aindicates no crises.Source: Authors’ calculations.

Finally, Figure 1.3 shows the effect of crises on recessions. It can be seenthat recessions with crises have a much higher loss of GDP than recessionswithout crises. This was particularly true in the interwar period. Also theaverage recovery time is somewhat higher in recessions with crises rather thanrecessions without crises.

In summary, the analysis of Bordo et al. (2000, 2001) leads to a numberof conclusions. Banking crises, currency crises, and twin crises have occurredunder a variety of different monetary and regulatory regimes. Over the last120 years crises have been followed by economic downturns lasting on averagefrom 2 to 3 years and costing 5 to 10 percent of GDP. Twin crises are associatedwith particularly large output losses. Recessions with crises were more severethan recessions without them.

The Bretton Woods period from 1945 to 1971 was quite special. Countrieseither regulated bank balance sheets to prevent them from taking very muchrisk or owned them directly to achieve the same aim. These measures weresuccessful in that there were no banking crises during this time and only onetwin crisis.

The interwar periodwas also special. Banking crises and currency criseswerewidespread. Moreover the output losses from these were severe particularlywhen they occurred together and there was a twin crisis.

Themost recent period does indeed appearmore crisis prone than any otherperiod except for the interwar years. In particular, it seems more crisis pronethan the Gold Standard Era, which was the last time that capital markets wereas globalized as they are now.


30.0

25.0

20.0

15.0

Per

cent

age

GD

P lo

ssYe

ars

10.0

5.0

0.0

5.0

4.0

3.0

2.0

1.0

0.0

1880–1913 1919–1939 1945–1971 1973–1997(21 nations)

1973–1997(56 nations)

1880–1913 1919–1939 1945–1971 1973–1997(21 nations)

1973–1997(56 nations)

GDP loss with

Average recovery time with

and without

and without

crises

crises

Figure 1.3. Recessions with and without crises (from Figure 2 of Bordo et al. 2001).

1.6 SOME RECENT CRISES

Now that we have seen a comparison of recent crises with crises in other eras,it is perhaps helpful to consider some of the more recent ones in greater detail.We start with those that occurred in Scandinavia in the early 1990’s.

1.6.1 The Scandinavian crises

Norway, Finland and Sweden experienced a classic boom–bust cycle that led totwin crises (see Heiskanen 1993 and Englund and Vihriälä 2006). In Norwaylending increased by 40 percent in 1985 and 1986. Asset prices soared whileinvestment and consumption also increased significantly. The collapse in oil

1.6 Some Recent Crises 15

prices helped burst the bubble and caused the most severe banking crisis andrecession since the war. In Finland an expansionary budget in 1987 resulted inmassive credit expansion. Housing prices rose by a total of 68 percent in 1987and 1988. In 1989 the central bank increased interest rates and imposed reserverequirements to moderate credit expansion. In 1990 and 1991 the economicsituation was exacerbated by a fall in trade with the Soviet Union. Asset pricescollapsed, banks had to be supported by the government and GDP shrank by7 percent. In Sweden a steady credit expansion through the late 1980’s led to aproperty boom. In the fall of 1990 credit was tightened and interest rates rose.In 1991 a number of banks had severe difficulties because of lending based oninflated asset values. The government had to intervene and a severe recessionfollowed.

1.6.2 Japan

In the 1980’s the Japanese real estate and stock markets were affected by abubble. Financial liberalization throughout the 1980’s and the desire to supportthe United States dollar in the latter part of the decade led to an expansion incredit. During most of the 1980’s asset prices rose steadily, eventually reachingvery high levels. For example, the Nikkei 225 index was around 10,000 in1985. On December 19, 1989 it reached a peak of 38,916. A new Governorof the Bank of Japan, less concerned with supporting the US dollar and moreconcerned with fighting inflation, tightened monetary policy and this led toa sharp increase in interest rates in early 1990 (see Frankel 1993; Tschoegl1993). The bubble burst. The Nikkei 225 fell sharply during the first partof the year and by October 1, 1990 it had sunk to 20,222. Real estate pricesfollowed a similar pattern. The next few years were marked by defaults andretrenchment in the financial system. Three big banks and one of the largestfour securities firms failed. The real economy was adversely affected by theaftermath of the bubble and growth rates during the 1990’s and 2000’s havemostly been slightly positive or negative, in contrast to most of the post-warperiod when they were much higher. Using the average growth rate of GDP of4 percent from 1976–1991, the difference between trend GDP and actual GDPfrom 1992–1998 is around ¥340 trillion or about 68 percent of GDP (Mikitaniand Posen 2000, p. 32).

1.6.3 The Asian crisis

From the early 1950’s until the eve of the crisis in 1997 the “Dragons” (HongKong, Singapore, South Korea, and Taiwan) and the “Tigers” (Indonesia,Malaysia, the Philippines, and Thailand) were held up as models of successful


economic development. Their economies grew at high rates for many years.After sustained pressure, the Thai central bank stopped defending the baht onJuly 2, 1997 and it fell 14 percent in the onshore market and 19 percent inthe offshore market (Fourçans and Franck 2003, Chapter 10). This marked thestart of the Asian financial crisis.

The next currencies to come under pressure were the Philippine peso andthe Malaysian ringitt. The Philippine central bank tried to defend the peso byraising interest rates but it nevertheless lost $1.5 billion of foreign reserves. OnJuly 11 it let the peso float and it promptly fell 11.5 percent. The Malaysiancentral bank also defended the ringitt until July 11 before letting it float. TheIndonesian central bank defended the rupee until August 14.

The Dragons were also affected. At the beginning of August, Singaporedecided not to defend its currency and by the end of September it had fallen8 percent. Taiwan also decided to let their currency depreciate and were notmuch affected. Hong Kong’s exchange rate, which was pegged to the dollarcame under attack. However, it was able to maintain the peg. Initially theSouth Korean won appreciated against the other South East Asian currencies.However, in November it lost 25 percent of its value. By the end of December1997 which marked the end of the crisis the dollar had appreciated by 52, 52,78, 107, and 151 percent against theMalaysian, Philippine, Thai, South Korean,and Indonesian currencies, respectively.

Although the turbulence in the currencymarketswas over by the endof 1997,the real effects of the crisis continued to be felt throughout the region. Manyfinancial institutions, and industrial and commercial firms went bankrupt andoutput fell sharply. Overall, the crisis was extremely painful for the economiesinvolved.

1.6.4 The Russian crisis and long term capitalmanagement (LTCM)

In 1994 John Meriwether who had previously worked for Salomon Brothersand had been a very successful bond trader founded LTCM. In addition toMeriwether, the other partners included two Nobel-prize winning economists,Myron Scholes and Robert Merton, and a former vice-chairman of the FederalReserve Board, David Mullins. The fund had no problem raising $1.3 billioninitially (see http://www.erisk.com/Learning/CaseStudies/ref_case_ltcm.aspand Lowenstein 2000).

The fund’s main strategy was to make convergence trades. These involvedfinding securities whose returns were highly correlated but whose prices were

http://www.erisk.com/Learning/CaseStudies/ref_case_ltcm.asp

1.6 Some Recent Crises 17

slightly different. The fund would then short (i.e. borrow) the one with thehigh price and use the proceeds to go long in the one with the low price.The convergence trades that were taken included the sovereign bonds ofEuropean countries that were moving towards European Monetary Union,and on-the-run and off-the-run US government bonds. Since the price dif-ferences were small the strategy involved a large amount of borrowing. Forexample, at the beginning of 1998 the firm had equity of about $5 billion andhad borrowed over $125 billion.

In the first two years the fund was extremely successful and earned returnsfor its investors of around 40 percent. However, 1997 was not as successful witha return of 27 percent which was about the same as the return on equities thatyear. By this time LTCM had about $7 billion under management. Meriwetherdecided to return about $2.7 billion to investors as they were not able to earnhigh returns with so much money under management.

On August 17, 1998 Russia devalued the rouble and declared a moratoriumon about 281 billion roubles ($13.5 billion) of government debt. Despite thesmall scale of the default, this triggered a global crisis with extreme volatilityin many financial markets. Many of the convergence trades that LTCM hadmade started to lose money as the flight to quality caused prices to move inunexpected directions. By September 22, 1998 the value of LTCM’s capital hadfallen to $600 million. Goldman Sachs, AIG, and Warren Buffet offered to pay$250 million to buy out the partners and to inject $4 billion into the businessso that it would not be forced to sell out its positions. Eventually the FederalReserve Bank of New York coordinated a rescue whereby the banks that hadlent significant amounts to LTCM would pay $3.5 million for 90 percent ofthe equity of the fund and take over the management of the portfolio. Thereason the Fed did this was to avoid the possibility of a meltdown in globalasset markets and the systemic crisis that would follow.

1.6.5 The Argentina crisis of 2001–2002

During the 1970’s and 1980’s Argentina’s economy did poorly. It had a numberof inflationary episodes and crises. In 1991 it introduced a currency board thatpegged the Argentinian peso at a one-to-one exchange rate with the dollar.This ushered in a period of low inflation and economic growth. Despite thesefavorable developments, a number of weaknesses developed during this periodincluding an increase in public sector debt and a low share of exports inoutput and a high concentration of these in a limited number of sectors (seeIMF 2003).


In the last half of 1998 a number of events including the crisis in Brazil andthe resulting devaluation and the Russian crisis triggered a sharp downturnin Argentina’s economy. The public debt the government had accumulatedlimited the amount of fiscal stimulation that the government could undertake.Also the currency board meant that monetary policy could not be used tostimulate the economy. The recession continued to deepen. At the end of 2001,it began to become clearer that Argentina’s situation was not sustainable. Thegovernment tried to take a number of measures to improve the situation suchas modifying the way that the currency board operated. Exporters were subjectto an exchange rate that was subsidized and importers paid a tax. The effect ofthese kinds of measures was to lower confidence rather than raise it. Despitean agreement with the IMF in September 2001 to inject funds of $5 billionimmediately and the prospect of another $3 billion subsequently the situationcontinued to worsen. There were a number of attempts to restructure thepublic debt but again this did not restore confidence.

During November 28–30 there was a run on private sector deposits. Thegovernment then suspended convertibility in the sense that it imposed a num-ber of controls including a weekly limit of 250 pesos on the amount thatcould be withdrawn from banks. In December 2001, the economy collapsed.Industrial production fell 18 percent year-on-year. Imports fell by 50 per-cent and construction fell 36 percent. In January 2002 the fifth presidentin three weeks introduced a new currency system. This involved multipleexchange rates depending on the type of transaction. In February this wasabolished and the peso was allowed to float and it soon fell to 1.8 pesos to thedollar.

Overall the crisis was devastating. Real GDP fell by about 11 percent in2002 and inflation in April 2002 went to 10 percent a month. The governmentdefaulted on its debt. Although the economy started to recover in 2003 andhas done well since then, it will be some time before it retains its pre-crisisactivity.

1.7 THE COSTS OF CRISES

There is a large literature on the costs of crises and their resolution (see, e.g.Bordo et al. 2001; Hoggarth et al. 2002; Roubini and Setser 2004; Boyd et al.2005; andHonohan and Laeven 2005).Much of the debate has been concernedwith how exactly tomeasure costs.A large part of the early literature focused onthe fiscal costs. This is the amount that it costs the government to recapitalizebanks and reimburse insured depositors. However, these are mostly transfers

1.8 Theories of Crises 19

rather than true costs. The subsequent literature has focused more on the lostoutput relative to a benchmark such as trend growth rate.

There are two important aspects of the costs of crises. The first is the highaverage cost and the second is the large variation in the amount of costs.Boyd et al. (2005) estimate the average discounted present value of losses ina number of different ways. Depending on the method used the mean loss isbetween 63 percent and 302 percent of real per capita GDP in the year beforethe crisis starts. The distribution of losses is very large. In Canada, France,Germany, and the US, which experienced mild nonsystemic crises, there wasnot any significant slowdown in growth and costs were insignificant. How-ever, at the other extreme the slowdown and discounted loss in output wereextremely high. In Hong Kong the discounted PV of losses was 1,041 percentof real output the year before the crisis.

It is the large average costs and the very high tail costs of crises that makespolicymakers so averse to crises. This is why in most cases they go to such greatlengths to avoid them. However, it is not clear that this is optimal. There aresignificant costs associated with regulations to avoid crises and in many casescrises are not very costly. An important theme of this book is that the costs ofavoiding crises must be traded off against the costs of allowing crises.

1.8 THEORIES OF CRISES

The contrast between the majority view concerning the cause of crises in the1930’s and the view of many today is striking. In the 1930’s the market was theproblem and government intervention through regulation or direct ownershipof banks was the solution. Today many argue that inconsistent governmentmacroeconomic policies or moral hazard in the financial system caused bygovernment guarantees is at the root of recent crises. Here the view is thatgovernment is the cause of crises and not the solution. Market forces are thesolution.

In this book we aim to provide some perspective on this debate by develop-ing a theoretical approach to analyze financial crises. In each chapter we willdevelop the basic ideas and then provide a brief account of the theoretical andempirical literature on the topic.

We start in Chapter 2 with some background material. In particular, wereview the basics of time, uncertainty, and liquidity. For many readers whoare quite familiar with this material it will be better to proceed straight toChapter 3. For those who are not, or who need a refresher on models ofresource allocation over time and with uncertainty, Chapter 2 will provide


an introduction. The first part of the chapter develops basic ideas related toconsumption and saving, and production, such as dated commodities andforward markets. The second part considers uncertainty and introduces statesof nature, contingent commodities, complete markets, and Arrow securities.Attitudes toward risk, and the roles of insurance and risk pooling are alsointroduced. The final part of the chapter considers how liquidity and liquiditypreference can be modeled.

Chapter 3 considers intermediation. In order to understand how bank-ing crises arise it is first necessary to develop a theory of banking or moregenerally of intermediation. The approach adopted is to model intermedi-aries as providing liquidity insurance to consumers. Using this foundation twoapproaches to crises can be developed. Both views of crises have a long history.One view, well expounded by Kindleberger (1978), is that they occur spon-taneously as a panic. The modern version was developed by Bryant (1980) andDiamond andDybvig (1983). The analysis is based on the existence of multipleequilibria. In at least one equilibrium there is a panic while in another thereis not.

The business cycle theory also has a long history (see, e.g. Mitchell 1941).The basic idea is that when the economy goes into a recession or depressionthe returns on bank assets will be low. Given their fixed liabilities in the formof deposits or bonds they may unable to remain solvent. This may precipitatea run on banks. Gorton (1988) showed empirically that in the US in the latenineteenth and early twentieth centuries, a leading economic indicator basedon the liabilities of failed businesses could accurately predict the occurrenceof banking crises. The second part of Chapter 3 develops this approach tocrises.

One of the most important causes of crises is a dramatic collapse in assetprices. One explanation for this drop in prices, which is the basis for thebusiness cycle view of crises examined in Chapter 3, is that expected futurecash flows fall. Another possibility is that prices are low because of a shortageof liquidity. Chapter 4 investigates the operation of asset markets where assetprice volatility is driven by liquidity shocks. The model is similar to that inChapter 3, except there are no intermediaries. In addition there is a fixed costto participating in markets and this can lead to limited market participation.When liquidity is plentiful, asset prices are driven by expected future payoffsin the usual way. However, when there is a scarcity of liquidity there is “cash-in-the-market pricing.” In this case, an asset’s price is simply the ratio of theamount sold to the amount of cash or liquidity that buyers have. Ex post buyerswould like to have more liquidity when there is cash-in-the-market pricing.Ex ante they balance the opportunity cost of holding liquidity when liquidity isplentiful against the profits to be made when liquidity is scarce. This theory of

1.8 Theories of Crises 21

asset price determination is consistent with significant asset price volatility. It isshown that there can bemultiple Pareto-ranked equilibria. In one equilibrium,there is limited participation and asset prices are volatile. In another, which isPareto-preferred, there is complete participation and asset prices are not veryvolatile.

Although in some crises the initial trigger is a large shock, in others it appearsthe trigger is a small event. For example, in the Russian crisis of 1998 discussedabove, the moratorium on debt payments that triggered the crisis involved atiny proportion of the world’s assets. Nevertheless it had a huge impact onthe world’s financial markets. There was subsequently a period of extremeturbulence in financial markets. Understanding how this type of financialfragility can arise is the topic of Chapter 5. Rather than just focusing on banksas inChapter 3, or onmarkets as inChapter 4, here the interaction of banks andmarkets is considered. The markets are institutional markets in the sense thatthey are for banks and intermediaries to share risks and liquidity. Individualscannot directly access these markets but instead invest their funds in banksthat have access to the markets. As in Chapter 4, the key to understanding theform of equilibrium is the incentives for providing liquidity to the market. Inorder for banks to be willing to hold liquidity, the opportunity cost of doingthis in states where the liquidity is not used must be balanced by the profits tobe made when liquidity is scarce and there is cash-in-the-market pricing. It ispossible to show that if such events are rare then very large changes in pricescan be triggered by small changes in liquidity demand. These price changescan cause bankruptcy and disruption. There is financial fragility.

While in Chapters 3–5 the focus is on understanding the positive aspectsof how various types of crisis can arise, in Chapter 6 we develop a generalframework for understanding the normative aspects of crises. The model is abenchmark for investigating the welfare properties of financial systems. Simi-larly toChapter 5, there are both intermediaries andmarkets.However,whereasin Chapter 5 markets were incomplete in the sense that hedging opportunitieswere limited, here we assume financial markets are complete. In particular,it is possible for intermediaries to hedge all aggregate risks in the financialmarkets. Under these ideal circumstances it can be shown that Adam Smith’sinvisible hand works. The allocation of resources is efficient in the followingsense. If the contracts between intermediaries and consumers are completein that they can also be conditioned on aggregate risks, then the allocation is(incentive) efficient.

Many contracts observed in practice between intermediaries and consumerssuch as debt and deposit contracts are incomplete. Provided financial marketsare complete, then even if contracts between intermediaries and consumersare incomplete, it can be shown the allocation is constrained efficient. In other


words, a planner subject to the same constraints in terms of incomplete con-tracts with consumers could not do better. What is more it is shown thatthe equilibrium with incomplete contracts often involves there being financialcrises. For example, if a bank uses a deposit contract then there can be a bank-ing crisis. This demonstrates that crises are not everywhere and always bad.In some cases they can increase effective contingencies and improve the alloca-tion of resources. Of course, we are not saying that crises are always good, onlythat in some cases they can be, in particular when financial markets are com-plete and contracts between intermediaries and consumers are incomplete. Iffinancialmarkets are incomplete then crises can indeed be bad. For example, asmentioned the financial fragility considered in Chapter 5 occurs because mar-kets are incomplete. Thus the contribution of Chapter 6 is to identify whenthere are market failures that potentially lead to a loss of welfare.

Having identified when there is a market failure, the natural question thatfollows is whether there exist policies that can correct the undesirable effectsof such failures. This is the topic of Chapter 7. Two types of regulation areconsidered. The first is the regulation of bank capital and the second is the regu-lation of bank liquidity. Simple examples with constant relative risk aversionconsumers are analyzed when financial markets are incomplete. It is shownthat the effect of bank and liquidity regulation depend critically on the degreeof relative risk aversion.When relative risk aversion is sufficiently low (below 2)increasing levels of bank capital above what banks would voluntarily hold canmake everybody better off. For bank liquidity regulation, requiring banks tohold more liquidity than they would choose to is welfare improving if relativerisk aversion is above 1. The informational requirements for these kinds ofintervention are high. Thus it may be difficult to improve welfare throughthese kinds of regulation as a practical matter.

The analysis in Chapters 6 and 7 stresses the ability of investors to sharedifferent risks. Risk sharing to the extent it is possible occurs because of explicitcontingencies in contracts or effective contingencies that can occur if there isdefault. Liquidity is associated with supplies of the consumption good. Therehas been no role for money or variations in the price level. Chapter 8 considersthe effect of allowing for money and the denomination of debt and othercontracts in nominal terms. It is shown that if the central bank can vary theprice level then this provides another way for risk to be shared. This is true forrisks shared within a country. It is also true for risks shared between countries.By varying the exchange rate appropriately a central bank can ensure risk isshared optimally with the rest of the world. However, such international risksharing creates a moral hazard because of the possibility that a country willborrow a lot in domestic currency and then expropriate the lenders by inflatingaway the value of the currency.

1.9 Concluding Remarks 23

The final two chapters in the book consider two forms of crisis that appearto be particularly important but which were not considered earlier. In manyinstances financial crises occur after a bubble in asset prices collapses. Howthese bubbles form and collapse and their effect on the financial system is thesubject of Chapter 9. The most important recent example of this phenomenonis Japan which was discussed above. In the mid 1980’s the Nikkei stock indexwas around 10,000. By the end of the decade it had risen to around 40,000.A new governor of the Bank of Japanwhowas concerned that a loosemonetarypolicy had kindled prospects of inflation decided to increase interest ratessubstantially. This pricked the bubble and caused stock prices to fall. Within afew months they had fallen by half. Real estate prices continued to rise for overa year however they then also started to fall. Fifteen years later both asset pricesand real estate are significantly lower with stocks and real estate at around aquarter of their peak value. The fall in asset prices has led to a fall in growth anda banking crisis. Japan is by no means the only example of this phenomenon.It can be argued the Asian crisis falls into this category. In the US the Roaring1920’s and the Great Depression of the 1930’s are another example.

TheAsian crisis illustrated another important phenomenon, contagion. Theepisode started in Thailand and spread to many other countries in the regionincluding South Korea, Malaysia, Indonesia, Hong Kong, the Philippines andSingapore. Interestingly it did not affect Taiwan nearly as much. Other regions,particularly South America, were also affected. Understanding the contagiousnature of many crises has become an important topic in the literature. Thereare a number of theories of contagion. One is based on trade and real links,another is based on interbank markets, another on financial markets and oneon payments systems. Contagion through interbank markets is the subjectmatter of Chapter 10.

1.9 CONCLUDING REMARKS

Theword crisis is used inmanydifferentways. This naturally raises the questionof when a situation is a crisis and when it is not. It is perhaps helpful toconsider the definition of crises. According to the dictionary (dictionary.com)a crisis is:

1. (a) the turning point for better or worse in an acute disease or fever(b) a paroxysmal attack of pain, distress, or disordered function(c) an emotionally significant event or radical change of status in a

person’s life


2. the decisive moment (as in a literary plot)3. (a) an unstable or crucial time or state of affairs inwhich a decisive change

is impending; especially : one with the distinct possibility of a highlyundesirable outcome

(b) a situation that has reached a critical phase.

This gives a range of the senses in which the word is used in general. Withregard to financial crises it is also used in a wide range of situations. Bankingcrises usually refer to situations wheremany banks simultaneously come underpressure and may be forced to default. Currency crises occur when there arelarge volumes of trade in the foreign exchange market which can lead to adevaluation or revaluation. Similarly it is used in many other situations wherebig changes, usually bad, appear possible. This is the sense in which we areusing the word in this book.

Historically, the study of financial crises was an important field in eco-nomics. The elimination of banking crises in the post-war period significantlyreduced interest in crises and it became an area for economic historians. Nowthat crises have reemerged much work remains to be done using modern the-oretical tools to understand the many aspects of crises. This book is designedto give a brief introduction to some of the theories that have been used to tryand understand these complex events.

There is a significant empirical literature on financial crises. Much of thiswork is concerned with documenting regularities in the data. Since the theoryis at a relatively early stage there is relatively little work trying to distinguishbetween different theories of crises. In the chapters below the historical andempirical work is discussed as a background to the theory.Much work remainsto be done in this area too.

There is a tendency in much of the literature on crises to argue that theparticular theory being presented is “THE” theory of crises. As even the briefdiscussion in this chapter indicates crises are complex phenomena in practice.One of the main themes of this book is that there is no one theory of crises thatcan explain all aspects of the phenomena of interest. In general, the theoriesof crises that we will focus on are not mutually exclusive. Actual crises maycontain elements of some combination of these theories.

REFERENCES

Bagehot, W. (1873). Lombard Street: A Description of the Money Market, London:H. S. King.

Bordo, M. (1986). “Financial Crises, Banking Crises, Stock Market Crashes and theMoney Supply: Some International Evidence, 1870-1933,” in F. Capie and G. Wood

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(eds.), Financial Crises and the World Banking System. New York: St. Martin’s Press,1986, 190–248.

Bordo, M. and B. Eichengreen (1999). “Is Our Current International EconomicEnvironment Unusually Crisis Prone?”Working paper, Rutgers University.

Bordo, M., B. Eichengreen, D. Klingebiel and M. Martinez-Peria (2000). “Is the CrisisProblem Growing More Severe?”Working paper, University of California, Berkeley.Updated version of previous paper. http://emlab.berkeley.edu/users/eichengr/research.html – this link also allows you to download the figures: scrolldown the page until you reach the link to the paper and figures – see alsohttp://www.haas.berkeley.edu/ arose/BEKSc.pdf

Bordo, M., B. Eichengreen, D. Klingebiel and M. Martinez-Peria (2001). “Is the Cri-sis Problem Growing More Severe?” Economic Policy, April 2001, 53–82 + WebAppendix.

Boyd, J., S. Kwak, and B. Smith (2005). “The Real Output Losses Associated withModern Banking Crises,” Journal of Money, Credit, and Banking 37, 977–999.

Bryant, J. (1980). “A Model of Reserves, Bank Runs, and Deposit Insurance,” Journal ofBanking and Finance 4, 335–344.

Diamond, D. and P. Dybvig (1983). “Bank Runs, Deposit Insurance, and Liquidity,”Journal of Political Economy 91, 401–419.

Englund, P. and V. Vihriälä (2006). “Financial Crises in Developed Economies: TheCases of Finland and Sweden,”Chapter 3 in Lars Jonung (ed.),Crises,MacroeconomicPerformance and Economic Policies in Finland and Sweden in the 1990s: AComparativeApproach, forthcoming.

Fourçans, A. and R. Franck (2003). Currency Crises: A Theoretical and EmpiricalPerspective, Northampton, MA: Edward Elgar.

Frankel, J. (1993). “The Japanese Financial System and the Cost of Capital,” in S. Takagi(ed.), Japanese Capital Markets: New Developments in Regulations and Institutions,Oxford: Blackwell, 21–77.

Gorton, G. (1988). “Banking Panics and Business Cycles,”Oxford Economic Papers 40,751–781.

Heiskanen,R. (1993).“The BankingCrisis in theNordic Countries,”Kansallis EconomicReview 2, 13–19.

Hoggarth, G., R. Reis, and V. Saporta (2002). “Costs of Banking System Instability:Some Empirical Evidence,” Journal of Banking and Finance 26, 825–855.

Honohan, P. and L. Laeven (2005). Systemic Financial Crises: Containment andResolution, Cambridge, UK: Cambridge University Press.

IMF (2003). “Lessons form the Crisis in Argentina,” Washington, DC: IMF,http://www.imf.org/external/np/pdr/lessons/100803.pdf.

Kindleberger, C. (1978). Manias, Panics, and Crashes: A History of Financial Crises,New York: Basic Books.

Kindleberger, C. (1993). A Financial History of Western Europe (second edition),New York: Oxford University Press.

Lowenstein, R. (2000). When Genius Failed: The Rise and Fall of Long-Term CapitalManagement, New York: Random House.

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Mikitani, R. and A. Posen (2000). Japan’s Financial Crisis and its Parallels toU.S. Experience, Washington, DC: Institute for International Economics, SpecialReport 13.

Mitchell,W. (1941).Business Cycles andTheir Causes, Berkeley: University of CaliforniaPress.

Roubini, N. and B. Setser (2004). Bailouts or Bail-Ins? Responding to Financial Crises inEmerging Economies, Washington, DC: Institute for International Economics.

Sprague, O. (1910). A History of Crises Under the National Banking System, NationalMonetary Commission,Washington DC: U.S. Government Printing Office.

Studenski, P. and H. Krooss (1963). Financial History of the United States (secondedition), New York: McGraw Hill.

Timberlake, R. (1978). The Origins of Central Banking in the United States, CambridgeMA: Harvard University Press.

Tschoegl, A. (1993). “Modeling the Behaviour of Japanese Stock Indices,” in S. Takagi(ed.), Japanese Capital Markets: New Developments in Regulations and Institutions,Oxford: Blackwell, 371–400.

Wilson, J., R. Sylla and C. Jones (1990). “Financial Market Panics and Volatility in theLong Run, 1830–1988,” in E. White (ed.), Crashes and Panics, Illinois: Dow-JonesIrwin, 85–125.

2

Time, uncertainty, and liquidity

Financial economics deals with the allocation of resources over time and inthe face of uncertainty. Although we use terms like “present values,” “statesof nature,” and “contingent commodities” to analyze resource allocation inthese settings, the basic ideas are identical to those used in the analysis ofconsumer and producer behavior in ordinary microeconomic theory. In thischapter we review familiar concepts such as preferences, budget constraints,and production technologies in a new setting, where we use them to study theintertemporal allocation of resources and the allocation of risk. We use simpleexamples to explain these ideas and later show how the ideas can be extendedand generalized.

2.1 EFFICIENT ALLOCATION OVER TIME

We begin with the allocation of resources over time. Although we introducesome new terminology, the key concepts are the same as concepts familiarfrom the study of efficient allocation in a “timeless” environment. We assumethat time is divided into two periods, which we can think of as representingthe “present” and the “future.”We call these periods dates and index them byt = 0, 1, where date 0 is the present and date 1 is the future.

2.1.1 Consumption and saving

Suppose a consumer has an income stream consisting of Y0 units of a homo-geneous consumption good at date 0 and Y1 units of the consumption goodat date 1. The consumer’s utility U (C0,C1) is a function of his consumptionstream (C0,C1), where C0 is consumption at date 0 and C1 is consumptionat date 1. The consumer wants to maximize his utility but first has to decidewhich consumption streams (C0,C1) belong to his budget set, that is, whichstreams are feasible for him. There are several ways of looking at this question.They all lead to the same answer, but it is worth considering each one in turn.

28 Chapter 2. Time, Uncertainty, and Liquidity

Borrowing and lending

One way of posing the question (of which consumption streams the consumercan afford) is to ask whether the income stream (Y0,Y1) can be transformedinto a consumption stream (C0,C1) by borrowing and lending. For simplicity,we suppose there is a bank that is willing to lend any amount at the fixedinterest rate i > 0 per period, that is, the bank will lend one unit of presentconsumption today in exchange for repayment of (1 + i) units in the future.Suppose the consumer decided to spendC0 > Y0 today. Then hewould have toborrow B = C0 − Y0 in order to balance his budget today, and this borrowingwould have to be repaid with interest iB in the future. The consumer couldafford to do this if and only if his future income exceeds his future consumptionby the amount of the principal and interest, that is,

(1 + i)B ≤ Y1 − C1.

We can rewrite this inequality in terms of the consumption and income streamsas follows:

C0 − Y0 ≤ 1

1 + i (Y1 − C1).

Conversely, if the consumer decided to consume C0 ≤ Y0 in the present, hecould save the difference S = Y0−C0 and deposit it with the bank.We supposethat the bank is willing to pay the same interest rate i > 0 on deposits thatit earns on loans, that is, one unit of present consumption deposited with thebank today will be worth (1+ i) units in the future. The consumer will receivehis savings with interest in the future, so his future consumption could exceedhis income by (1 + i)S, that is,

C1 − Y1 ≤ (1 + i)S.We can rewrite this inequality in terms of the consumption and income streamsas follows:

C0 − Y0 ≤ 1

1 + i (Y1 − C1).

Notice that this is the same inequality as we derived before. Thus, any feasibleconsumption stream,whether it involves saving or borrowing,must satisfy thesame constraint. We call this constraint the intertemporal budget constraintand write it for future reference in a slightly different form:

C0 + 1

1 + i C1 ≤ Y0 + 1

1 + i Y1. (2.1)

2.1 Efficient Allocation Over Time 29

C1

C0Y0+Y1/(1+i)

(1+i )Y0+Y1

Figure 2.1. Intertemporal budget constraint.

Figure 2.1 illustrates the set of consumption streams (C0,C1) that satisfythe intertemporal budget constraint. It is easy to see that the income stream(Y0,Y1) must satisfy the intertemporal budget constraint. If there is neitherborrowing nor lending in the first period then C0 = Y0 and C1 = Y1.The endpoints of the line represent the levels of consumption that would bepossible if the individual were to consume as much as possible in the presentand future, respectively. For example, if he wants to consume as much aspossible in the present, he has Y0 units of income today and he can borrowB = Y1/(1 + i) units of the good against his future income. This is the max-imum he can borrow because in the future he will have to repay the principalB plus the interest iB, for a total of (1 + i)B = Y1. So the maximum amounthe can spend today is given by

C0 = Y0 + B = Y0 + Y1

1 + i .

Conversely, if he wants to consume as much as possible in the future, he willsave his entire income in the present. In the future, he will get his savings withinterest (1 + i)Y0 plus his future income Y1. So the maximum amount he canspend in the future is

C1 = (1 + i)Y0 + Y1.

Suppose now that consumption in the first period is increased by �C0. Byhow much must future consumption be reduced? Every unit borrowed in thefirst period will cost (1 + i) in the second because interest must be paid. Sothe decrease in second period consumption is �C1 = −(1 + i)�C0. This


shows that he can afford any consumption stream on the line between the twoendpoints with constant slope = −(1 + i). (See Figure 2.1.)

We have shown that any consumption stream that can be achieved byborrowing and lendingmust satisfy the intertemporal budget constraint. Con-versely, we can show that any consumption stream (C0,C1) that satisfies theintertemporal budget constraint can be achieved by some feasible pattern ofborrowing or lending (saving). To see this, suppose that the intertemporalbudget constraint is satisfied by the consumption stream (C0,C1). If C0 > Y0

we assume the consumer borrowsB = C0−Y0. In the future he has to repay hisloan with interest, so he only has Y1 − (1 + i)B left to spend on consumption.However, the intertemporal budget constraint ensures that his planned futureconsumption C1 satisfies

C1 ≤ (1 + i)(Y0 − C0) + Y1

= Y1 − (1 + i)(C0 − Y0).

So the consumer can borrow B units today and repay it with interest tomorrowand still afford his planned future consumption. The other case whereC0 ≤ Y0

is handled similarly. Thus, we have seen that the income stream (Y0,Y1) canbe transformed into the consumption stream (C0,C1) through borrowing orlending at the interest rate i if and only if it satisfies the intertemporal budgetconstraint (2.1).

Wealth and present values

Another way of thinking about the set of affordable consumption streamsmakes use of the concept of present values. The present value of any good is theamount of present consumption that someone would give for it. The presentvalue of one unit of present consumption is 1. The present value of one unit offuture consumption is 1/(1+ i), because one unit of present consumption canbe converted into 1 + i units of future consumption, and vice versa, throughborrowing and lending. Thus, the present value of the income stream (Y0,Y1),that is, the value of (Y0,Y1) in terms of present consumption is

PV (Y0,Y1) ≡ Y0 + 1

1 + i Y1

and the present value of the consumption stream (C0,C1) is

PV (C0,C1) ≡ C0 + 1

1 + i C1.


The intertemporal budget constraint says that the present value of theconsumption stream (C0,C1) must be less than or equal to the present valueof the consumer’s income stream.

The present value of the income stream (Y0,Y1) is also called the consumer’swealth, denoted by W ≡ Y0 + 1

1+i Y1. The intertemporal budget constraintallows the consumer to choose any consumption stream (C1,C2)whosepresentvalue does not exceed his wealth, that is,

C0 + C1

1 + i ≤ W . (2.2)

Dated commodities and forward markets

There is still a third way to interpret the intertemporal budget constraint (2.1).We are familiar with the budget constraint of a consumer who has to dividehis income between two goods, beer and pizza, for example. There is a priceat which each good can be purchased and the value of consumption is calcu-lated by multiplying the quantity of each good by the price and adding thetwo expenditures. The consumer’s budget constraint says that the value of hisconsumption must be less than or equal to his income. The intertemporalbudget constraint (2.1) can be interpreted in this way too. Suppose we treatpresent consumption and future consumption as two different commoditiesand assume that there are markets on which the two commodities can betraded. We assume these markets are perfectly competitive, so the consumercan buy and sell as much as he likes of both commodities at the prevail-ing prices. The usual budget constraint requires the consumer to balance thevalue of his purchases and expenditures on the two commodities. If p0 is theprice of present consumption and p1 is the price of future consumption, thenthe ordinary budget constraint can be written as

p0C0 + p1C1 ≤ p0Y0 + p1Y1.

Suppose that we want to use the first-period consumption good as ournumeraire, that is, measure the value of every commodity in terms of thisgood. Then the price of present consumption is p0 = 1, since one unit of thegood at date 0 is worth exactly one (unit of the good at date 0). How much isthe good at date 1worth in terms of the good at date 1? If it is possible to borrowand lend at the interest rate i, the price of the good at date 1 will be determinedby arbitrage. If p1 > 1

1+i , then anyone can make a riskless arbitrage profit by

selling one unit of future consumption for p1, using the proceeds to buy 11+i

units of present consumption, and investing the 11+i units at the interest rate i


to yield (1+ i) 11+i = 1 unit of future consumption. This strategy yields a profit

of p1 − 11+i at date 0 and has no cost since the unit of future consumption

that is sold is provided by the investment at date 0. Such a risk free profit isincompatible with equilibrium, since anyone can use this arbitrage to generateunlimited wealth. Thus, in equilibrium we must have p1 ≤ 1

1+i .A similar argument can be used to show that if p1 < 1

1+i , it is possible to

make a risk free profit by borrowing 11+i units of present consumption, buying

one unit of future consumption at the price p1, and using it to repay the loanat date 1. Thus, in equilibrium, we must have p1 ≥ 1

1+i .Putting these two arbitrage arguments together, we can see that, if it is

possible to borrow and lend at the interest rate i and present consumptionis the numeraire, the only prices consistent with equilibrium are p0 = 1 andp1 = 1

1+i .Substituting the prices p0 = 1 and p1 = 1

1+i into the budget constraintabove,we see that it is exactly equivalent to the intertemporal budget constraint(2.1). Borrowing and lending at a constant interest rate is equivalent to havinga market in which present and future consumption can be exchanged at theconstant prices (p0, p1). The same good delivered at two different dates is twodifferent commodities and present and future consumption are, in fact, simplytwo different commodities with two distinct prices. From this point of view,the intertemporal budget constraint is just a new interpretation of the familiarconsumer’s budget constraint.

Consumption and saving

Since the consumer’s choices among consumption streams (C0,C1) are com-pletely characterized by the intertemporal budget constraint, the consumer’sdecision problem is to maximize his utilityU (C0,C1) by choosing a consump-tion stream (C0,C1) that satisfies the budget constraint. We represent thisdecision problem schematically by

max U (C0,C1)

s.t. C0 + C1

1 + i = W .

Note that here we assume the budget constraint is satisfied as an equality ratherthan an inequality. Sincemore consumption is preferred to less, there is no lossof generality in assuming that the consumer will always spend as much as hecan on consumption. The solution to this maximization problem is illustratedin Figure 2.2, where the optimum occurs at the point on the budget constraintwhere the indifference curve is tangent to the budget constraint.


C1

C0(W,0)

(C0,C1)

0

Figure 2.2. Consumption and saving.

The slopeof the budget constraint is−(1+i) and the slopeof the indifferencecurve at the optimum point is

−∂U∂C0

(C∗0 ,C∗

1 )

∂U∂C1

(C∗0 ,C∗

1 ),

so the tangency condition can be written as

∂U∂C0

(C∗0 ,C∗

1 )

∂U∂C1

(C∗0 ,C∗

1 )= (1 + i).

It is easy to interpret the first-order condition by rewriting it as

∂U

∂C0(C∗

0 ,C∗1 ) = (1 + i) ∂U

∂C1(C∗

0 ,C∗1 ).

The left hand side is the marginal utility of consumption at date 0. The righthand side is the marginal utility of (1+ i) units of consumption at date 1. Oneunit of the good at date 0 can be saved to provide 1+ i units of the good at date1. So the first-order condition says that the consumer is indifferent betweenconsuming one unit at date 0 and saving it until date 1 when it will be worth(1 + i) units and then consuming the (1 + i) units at date 1.

An alternative to the graphical method of finding the optimum is to use themethod of Lagrange, which requires us to form the Lagrangean function

L(C0,C1, λ) = U (C0,C1) − λ

(C0 + 1

1 + i C1 −W)


and maximize the value of this function with respect to C0, C1, and theLagrange multiplier λ. A necessary condition for a maximum at (C∗

0 ,C∗1 , λ∗)

is that the derivatives of L(C0,C1, λ) with respect to these variables should allbe zero. Then

∂L∂C0

(C∗0 ,C∗

1 , λ∗) = ∂U

∂C0

(C∗

0 ,C∗1

)− λ∗ = 0,

∂L∂C1

(C∗0 ,C∗

1 , λ∗) = ∂U

∂C0

(C∗

0 ,C∗1

)− λ∗

1 + i = 0,

∂L∂λ

(C∗0 ,C∗

1 , λ∗) = C∗0 + 1

1 + i C∗1 −W = 0.

The first two conditions are equivalent to the tangency condition derivedearlier. To see this, eliminate λ∗ from these equations to get

∂U

∂C0

(C∗

0 ,C∗1

) = λ∗ = (1 + i) ∂U

∂C0

(C∗

0 ,C∗1

).

The last of the three conditions simply asserts that the budget constraint mustbe satisfied.

As before, the optimum (C∗0 ,C∗

1 ) is determined by the tangency conditionand the budget constraint.

Clearly, the optimal consumption stream (C∗0 ,C∗

1 ) will be a function ofthe consumer’s wealth W and the rate of interest i. If the pattern of incomewere (W , 0) instead of (Y1,Y2) the value of wealth would be the same andhence the budget line would be the same. So the same point (C∗

1 ,C∗2 ) would

be chosen. In fact (Y1,Y2) could move to any other point on the budget linewithout affecting consumption. Only savings or borrowing would change.

On the other hand, an increase inW toW ′, say, will shift the budget line outand increase consumption. The case illustrated in Figure 2.3 has the specialproperty that the marginal rate of substitution is constant along a straight lineOA through the origin. The slope of the budget line does not change so inthis case the point of tangency moves along the line OA asW changes. In thisspecial case, C1 is proportional toW .

Problems1. An individual consumer has an income stream (Y0,Y1) and can borrow and

lend at the interest rate i. For each of the following data points, determinewhether the consumption stream (C0,C1) lieswithin the consumer’s budget


C1

C0(W,0)0

A

Figure 2.3. The effect of an increase in wealth.

set (i.e. whether it satisfies the intertemporal budget constraint).

(C0,C1) (Y0,Y1) (1 + i)(10, 25) (15, 15) 2(18, 11) (15, 15) 1.1(18, 11) (15, 15) 1.5(10, 25) (15, 15) 1.8

Draw a graph to illustrate your answer in each case.2. An individual consumer has an income stream (Y0,Y1) = (100, 50) and

can borrow and lend at the interest rate i = 0.11. His preferences arerepresented by the additively separable utility function

U (C0,C1) = log C0 + 0.9 log C1.

The marginal utility of consumption in period t is

d log CtdCt

= 1

Ct.

Write down the consumer’s intertemporal budget constraint and the first-order condition that must be satisfied by the optimal consumption stream.Use the first-order condition and the consumer’s intertemporal budgetconstraint to find the consumption stream (C∗

0 ,C∗1 ) that maximizes utility.

How much will the consumer save in the first period? How much will hissavings be worth in the second period? Check that he can afford the optimalconsumption C∗

1 in the second period.


2.1.2 Production

Just as we can cast the consumer’s intertemporal decision into the familiarframework of maximizing utility subject to a budget constraint,we can cast thefirm’s intertemporal decision into the form of a profit- or value-maximizationproblem.

Imagine a firm that can produce outputs of a homogeneous good in eitherperiod subject to a production technology with decreasing returns. Output atdate 0 is denoted by Y0 and output at date 1 is denoted by Y1. The possiblecombinations of Y0 and Y1 are described by the production possibility curveillustrated in Figure 2.4.

Y1

Y00

Figure 2.4. The production possibility curve.

Note the following properties of the production possibility curve:

• the curve is downward sloping to the right because the firm must reduceoutput tomorrow in order to increase the output today;

• the curve is convex upward because of the diminishing returns – as the firmdecreases output today, each additional unit of present output foregone addsless to output tomorrow.

The production technology can be represented by a transformation functionF(Y0,Y1). A pair of outputs (Y0,Y1) is feasible if and only if it satisfies theinequality

F(Y0,Y1) ≤ 0.

The function F is said to be increasing if an increase in Y0 or Y1 increases thevalue F(Y0,Y1). The function F is said to be convex if, for any output streams


(Y0,Y1) and (Y ′0,Y

′1) and any number 0 < t < 1,

F(t (Y0,Y1) + (1 − t ) (Y ′

0,Y′1

)) ≤ tF (Y0,Y1) + (1 − t ) F (Y ′0,Y

′1

).

If F is increasing, then the production possibility curve is downward sloping.If the function F is convex, the production possibility curve is convex upward.In other words, if (Y0,Y1) and (Y ′

0,Y′1) are feasible, then any point on the line

segment between them is feasible.To illustrate the meaning of the transformation curve, suppose that the

firm’s past investments produce an output of Y0 at the beginning of period0. The firm can re-invest K0 units and sell the remaining Y0 = Y0 − K0. Theinvestment of K0 units today produces Y1 = G(K0) in the future (assumethere is no investment in the future because the firm is being wound up). Thenthe firm can produce any combination of present and future goods (Y0,Y1)

for sale that satisfies Y0 ≤ Y0 and Y1 = G(Y0 − Y0). Then the transformationcurve F(Y0,Y1) can be defined by

F(Y0,Y1) = Y1 − G(Y0 − Y0).

Which combination of outputs Y0 and Y1 should the firm choose? In gen-eral, there may be many factors that will guide the firm’s decision, but undercertain circumstances the firm can ignore all these factors and consider onlythe market value of the firm. To see this, we need only recall our discussionof the consumer’s decision. Suppose that the firm is owned by a single share-holder who receives the firm’s outputs as income. If the consumer can borrowand lend as much as he wants at the rate i, all he cares about is his wealth, thepresent value of the income stream (Y0,Y1). The exact time-profile of income(Y0,Y1) does notmatter. So if the firmwants tomaximize its shareholder’s wel-fare, it should maximize the shareholder’s wealth. To make these ideas moreprecise, suppose that the firm has a single owner-manager who chooses boththe firm’s production plan (Y0,Y1) and his consumption stream (C0,C1) tomaximize his utility subject to his intertemporal budget constraint. Formally,we can write this decision problem as follows:

max U (C0,C1)

s.t. F(Y0,Y1) ≤ 0

C0 + 11+i C1 ≤ W ≡ Y0 + 1

1 + i Y1.

Then it is clear that the choice of (Y0,Y1) affects utility only through theintertemporal budget constraint and that anything that increases the presentvalue of the firm’s output stream will allow the consumer to reach a more


desirable consumption stream. Thus, the joint consumption and productiondecision above is equivalent to the following two-stage procedure. First, havethe firm maximize the present value of outputs:

max W ≡ Y0 + 1

1 + i Y1

s.t. F(Y0,Y1) ≤ 0.

The present value of outputs is also known as the market value of the firm, sothis operational rule can be rephrased to say that firms should alwaysmaximizemarket value. Then have the consumer maximize his utility taking the firm’smarket value as given:

max U (C0,C1)

s.t. C0 + 1

1 + i C1 ≤ W ,

whereW = Y0 + 11+i Y1. Figure 2.5 illustrates this principle for the case of a

single shareholder.

Y1

Y00

Figure 2.5. Value maximization and utility maximization.

In fact, this argument extends to the case where there are many shareholderswith different time preferences. Some shareholdersmay be impatient andwantto consume more in the present while others are more patient and are willingto postpone consumption, but all will agree that a change in production thatincreases the present value of output must be a good thing, because it increasesthe consumers’ wealth. Figure 2.6 illustrates the case of two shareholders withequal shares in the firm. Then they have identical budget constraints with


Y1

Y00

Figure 2.6. The separation theorem.

slope −(1 + i), i.e. their budget constraint is parallel to the maximum valueline that is tangent to the production possibility frontier. Each shareholder willchoose to consume the bundle of goods (C0,C1) that maximizes his utilitysubject to the budget constraint. Because they have different time preferences,represented here by different indifference curves, each shareholder will choosea different bundle of goods, as indicated by the different points of tangencybetween indifference curves and budget constraint. Nonetheless, both agreethat the firm should maximize its market value, because maximizing the valueof the firm has the effect of putting both shareholders on the highest poss-ible budget constraint. This is known as the separation theorem because thefirm’s decision to maximize its value is separate from shareholders’ decisionsto maximize their utility.

Problem3. A firm has 100 units of output at the beginning of period 0. It has three

projects that it can finance. Each project requires an input of I units ofthe good at date 0 and produces Y1 units of the good at date 1. The projectsare defined in the following table:

Project Investment I Output Y1

1 20 302 30 483 50 70

Which projects should the firm undertake when the interest factor is

1 + i = 2, 1.5, 1.1?


Trace the firm’s production possibility curve (the combinations of Y0 andY1 that are technologically feasible) assuming that the firm can undertakea fraction of a project. Use this diagram to illustrate how changes in thediscount factor change the firm’s decision.

2.2 UNCERTAINTY

In the same way that we can extend the traditional analysis of consumptionand production to study the allocation of resources over time, we can use thesame ideas to study the allocation of risk bearing under uncertainty. Onceagain we shall use a simple example to illustrate the general principles.

2.2.1 Contingent commodities and risk sharing

We assume that time is divided into two periods or dates indexed by t = 0, 1.At date 0 (the present) there is some uncertainty about the future. For example,an individual may be uncertain about his future income.We can represent thisuncertainty by saying that there are several possible states of nature. A stateof nature is a complete description of all the uncertain, exogenous factors thatmay be relevant for the outcome of an individual’s decision. For example, afarmer who is planting a crop may be uncertain about the weather. The sizeof his crop will depend on choices he makes about the time to plant, the useof fertilizers, etc., as well as the weather. In this case, we identify the weatherwith the state of nature. Each state would be a complete description of theweather – the amount of rainfall, the temperature, etc. – during the growingseason. The outcome of the farmer’s choices will depend on the parametershe determines – time to plant, etc. – and the state of nature. In other words,once we know the farmer’s choices and the state of nature, we should know thesize of the crop; but even after the farmer has determined all the parametersthat he controls, the fact that the state is still unknown means that the size ofthe crop is uncertain.

Inwhat followswe assume that the only uncertainty relates to an individual’sincome, so income is a function of the state of nature. The true state of natureis unknown at date 0 and will be revealed at date 1. For simplicity, suppose thatthere are two possible states indexed by s = H , L. The letters H and L standfor “high” and “low.”

Commodities are distinguished by their physical characteristics, by the dateat which they are delivered, and by the state in which they are delivered.Thus, the consumption good in state H is a different commodity from theconsumption good in state L. We call a good whose delivery is contingent

2.2 Uncertainty 41

on the occurrence of a particular state of nature a contingent commodity.By adopting this convention, we can represent uncertainty about income andconsumption in terms of a bundle of contingent commodities. If YH denotesfuture income in state H and YL denotes income in state L, the ordered pair(YH ,YL) completely describes the future uncertainty about the individual’sincome. By treating (YH ,YL) as a bundle of different (contingent) commod-ities, we can analyze choices under uncertainty in the same way that we analyzechoices among goods with different physical characteristics. We assume thatYH > YL , that is, income is higher in the “high” state.

An individual’s preferences over uncertain consumption outcomes can berepresented by a utility function that is defined on bundles of contingentcommodities. Let U (CH ,CL) denote the consumer’s utility from the bundleof contingent commodities (CH ,CL), where CH denotes future consumptionin state H and CL denotes future consumption in state L. Later, we introducethe notion of a state’s probability and distinguish between an individual’sprobability beliefs and his attitudes to risk. Here, an individual’s beliefs aboutthe probability of a state and his attitudes towards risk are subsumed in hispreferences over bundles of contingent commodities.

Complete markets

There are two equivalent ways of achieving an efficient allocation of risk. Oneapproach to the allocation of risk assumes that there are complete markets forcontingent commodities. An economy with complete markets is often referredto as an Arrow–Debreu economy. In an Arrow–Debreu economy, there is amarket for each contingent commodity and a prevailing price at which con-sumers can trade asmuch of the commodity as they like subject to their budgetconstraint. Let pH and pL denote the price of the contingent commodities cor-responding to statesH and L, respectively. The consumer’s income consists ofdifferent amounts of the two contingent commodities, YH units of the con-sumption good in stateH and YL units of the consumption good in state L.Wecan use the complete markets to value this uncertain income stream and theconsumer’s wealth is pHYH + pLYL . Then the consumer’s budget constraint,which says that his consumption expenditure must be less than or equal to hiswealth, can be written as

pHCH + pLCL ≤ pHYH + pLYL .

The consumer can afford any bundle of contingent commodities (CH ,CL) thatsatisfies this constraint. He chooses the consumption bundle that maximizes


his utility U (CH ,CL) subject to this constraint. We illustrate the consumer’sdecision problem in Figure 2.7.

CL

CH

(CH,CL)

0

Figure 2.7. Maximizing utility under uncertainty.

Arrow securities

The assumption of completemarkets guarantees efficient allocation of risk, butit may not be realistic to assume that every contingent commodity, of whichthere will be a huge number in practice, unlike in our example, can literally betraded at the initial date. Fortunately, there exists an alternative formulationwhich is equivalent in terms of its efficiency properties, but requires far fewermarkets. More precisely, it requires that securities and goods be traded on spotmarkets at each date, but the total number of spot markets is much less thanthe number of contingent commodities.

The alternative representation of the allocation of risk makes use of the ideaof Arrow securities. We define an Arrow security to be a promise to deliverone unit of money (or an abstract unit of account) if a given state occursand nothing otherwise. In terms of the present example, there are two typesof Arrow securities, corresponding to the states H and L respectively. Let qHdenote the price of the Arrow security corresponding to state H and let qLdenote the price of the Arrow security in state L. In other words, qH is the priceof one unit of account (one “dollar”) delivered in state H at date 1 and qL isthe price of one unit of account delivered in state L at date 1. A consumercan trade Arrow securities at date 0 in order to hedge against income risksat date 1. Let ZH and ZL denote the excess demand for the Arrow securities

2.2 Uncertainty 43

corresponding to states H and L respectively.1 If Zs > 0 then the consumer istaking a long position (offering to buy) in the Arrow security and if Zs < 0 theconsumer is taking a short position (offering to sell).We assume the consumerhas no income at date 0 – this period exists only to allow individuals to hedgerisks that occur at date 1 – so the consumer has to balance his budget by sellingone security in order to purchase the other. Suppose that the consumer choosesa portfolio Z = (ZH ,ZL) of Arrow securities. Then at date 1, once the truestate has been revealed, his budget constraint will be

pHCH ≤ pHYH + ZH , (2.3)

if state H occurs, and

pLCL ≤ pLYL + ZL , (2.4)

if state L occurs. The consumer will choose the portfolio Z and the consump-tion bundle (CH ,CL) to maximize U (CH ,CL) subject to the date-0 budgetconstraint

qHZH + qLZL ≤ 0

and the budget constraints (2.3) and (2.4). Since ZH = pH (CH − YH ) andZL = pL(CL − YL), the budget constraint at date 0 is equivalent to qH pH(CH − YH ) + qLpL(CL − YL) ≤ 0 or

qH pHCH + qLpLCL ≤ qH pHYH + qLpLYL .This looks just like the standard budget constraint in which we interpret pH =qH pH and pL = qLpL as the prices of contingent commodities and CH and CLas the demands for contingent commodities.

Now that we have seen how to interpret the allocation of risk in termsof contingent commodities, we can use the standard framework to analyzeefficient risk sharing. Figure 2.8 shows an Edgeworth box in which the axescorrespond to consumption in stateH and consumption in state L. A compet-itive equilibrium in which consumers maximize utility subject to their budgetconstraint leads to an efficient allocation of contingent commodities, that is,an efficient allocation of risk.

As usual, the conditions for efficiency include the equality of the two con-sumers’ marginal rates of substitution, but here the interpretation is different.

1 Each agent begins with a zero net supply of Arrow securities. Then agents issue securitiesfor one state in order to pay for their purchase of securities in the other. The vector Z representsthe agent’s net or excess demand of each security: a component Zs is negative if his supply of thesecurity (equals minus the excess demand) is positive and positive if his net demand is positive.


CL

CH0A

0B

Figure 2.8. Efficient allocation of risk between two agents.

We are equating the marginal rates of substitution between consumption inthe two states rather than two different (physical) goods. The marginal rate ofsubstitution will reflect an individual’s subjective belief about the probabilityof each state as well as his attitude toward risk.

2.2.2 Attitudes toward risk

To characterize individuals’ attitudes toward risk we introduce a special kindof utility function,which we call a vonNeumann–Morgenstern (VNM) utilityfunction. Whereas the standard utility function is defined on bundles of con-tingent commodities, theVNMutility is defined on quantities of consumptionin a particular state. Von Neumann and Morgenstern showed that, under cer-tain conditions, a rational individual would act so as to maximize the expectedvalue of his VNM utility function. If individuals satisfy the assumptions ofthe VNM theory, they will always make choices so as to maximize the valueof their expected utility. To see what this means in practice, let U (C) denotethe VNM utility of consuming C units of the good at date 1 and suppose thatthe probability of state s occurring is πs > 0 for s = H , L. Then the expectedutility of a consumption plan (CH ,CL) is πHU (CH )+πLU (CL). The decisionproblem of the consumer we encountered above can be re-written as

max πHU (CH ) + πLU (CL)

s.t. qHCH + qLCL ≤ qHYH + qLYL .

2.2 Uncertainty 45

The first-order conditions for this problem are

πsU′(Cs) = µqs ,

for s = H , L, where U ′(Cs) is the marginal utility of consumption in states and µ (the Lagrange multiplier associated with the budget constraint) can beinterpreted as the marginal utility of money. Notice that the marginal utilityof consumption in state s is multiplied by the probability of state s and it is theproduct – the expected marginal utility – which is proportional to the price ofconsumption in that state. Then the first-order condition can be interpreted assaying that the expected marginal utility of one unit of consumption in state sis equal to the marginal utility of its cost.

We typically think of individuals as being risk averse, that is, they avoid riskunless there is some advantage to be gained from accepting it. The clearest evi-dence for this property is the tendency to buy insurance. We can characterizerisk aversion and an individual’s attitudes to risk generally in the shape of theVNM utility function. Figure 2.9 shows the graph of a VNM utility function.Utility increases with income (the marginal utility of consumption is posi-tive) but the utility function becomes flatter as income increases (diminishingmarginal utility of consumption).

A VNM utility function has diminishing marginal utility of consumption ifand only if it is strictly concave. Formally, a VNM utility function U is strictlyconcave if, for any consumption levels C and C ′ (C �= C ′) and any number0 < t < 1, it satisfies the inequality

U(tC + (1 − t )C ′) > tU (C) + (1 − t )U (C ′). (2.5)

C0

U

U(C)

Figure 2.9. The von Neumann–Morgenstern utility function.


Concavity of theVNMutility function canbe interpreted as an attitude towardsrisk. To see this, suppose that an individual is offered a gamble in which hereceives C with probability t and C ′ with probability 1 − t . If his VNM utilityis strictly concave, he will prefer to receive the expected value tC + (1 − t )C ′for sure, rather than take the gamble. This is because the expected utility ofthe gamble (the right hand side of the inequality) is less than the utility of thesure thing. The utility function will always be strictly concave if the individualexhibits diminishingmarginal utility of income.An individualwho satisfies theassumption of diminishing marginal utility of income is said to be risk averse.(Draw the graph of a utility functionwith increasingmarginal utility of incomeand see how the comparison of the two options changes. An individual withthese preferences is called a risk lover.)

The conclusion then is that, faced with a choice between a risky incomedistribution and a degenerate distribution with the same expected value, a riskaverse individual will always choose the one without risk. In what follows, weassume that the VNM utility function is concave and, hence, individuals arerisk averse.

We have seen that risk aversion is associated with the curvature of the utilityfunction, in particular, with the fact that the marginal utility of income isdecreasing. Mathematically, this means that the second derivative of the utilityfunction U ′′(C) is less than or equal to zero. It would be tempting to take thesecond derivative U ′′(C) to be the measure of risk aversion. Unfortunately,the VNM utility function is only determined up to an affine transformation,that is, for any constants α and β > 0, the VNM utility function α + βU isequivalent to U in terms of the attitudes to risk that it implies. Thus, we mustlook for a measure that is independent of α and β > 0. Two such measures areavailable. One is known as the degree of absolute risk aversion

A(C) = −U′′(C)

U ′(C)

and the other is the degree of relative risk aversion

R(C) = −U′′(C)C

U ′(C).

There is a simple relationship between the degree of risk aversion and the riskpremium that an individual will demand to compensate for taking risk. Sup-pose that an individual has wealthW and is offered the following gamble.Withprobability 0.5 he wins a small amount h and with probability 0.5 he loses h.Since the expected value of the gamble is zero, the individual’s expected income

2.2 Uncertainty 47

is not changed by the gamble. Since a risk averse individual would rather haveW for sure than have an uncertain income with the same expected value,he will reject the gamble. The risk premium a is the amount he would have tobe given in order to accept the gamble. That is, a satisfies the equation

U (W ) = 1

2U (W + a − h) + 1

2U (W + a + h).

A Taylor expansion of the right hand side shows that, when h is small,

U (W ) ≈ U (W ) + U ′(W )a + 1

2U ′′(W )h2.

Thus,

a ≈ −U′′(W )

U ′(W )

h2

2= A(W )

h2

2.

So the risk premium a is equal to the degree of absolute risk aversion timesone-half the variance of the gamble (a measure of the risk). A similar inter-pretation can be given for the degree of relative risk aversion when the gambleconsists of winnings of hW and −hW with equal probability.

If the degree of absolute risk aversion is constant, the VNM utility functionmust have the form

U (C) = −e−AC ,

and A > 0 is the degree of absolute risk aversion. If the degree of relative riskaversion is constant and different from 1, then

U (C) = 1

1 − σC1−σ ,

where σ > 0 is the degree of relative risk aversion. This formula is not definedwhen the degree of relative risk aversion is σ = 1; however, the limiting valueof the utility function as σ → 1 is well defined and given by

U (C) = lnC ,

where lnC denotes the natural logarithm of C .The higher the degree of (relative or absolute) risk aversion, the more risk

averse the individual with the VNM utility function U (C) is.


2.2.3 Insurance and risk pooling

Returning to the example of efficient risk sharing studied earlier, we canuse the assumption that consumers have VNM utility functions to charac-terize the efficient risk sharing allocation more precisely. Suppose that thereare two individualsA andBwithVNMutility functionsUA andUB and incomedistributions (YAH ,YAL) and (YBH ,YBL) respectively. If the efficient allocationof consumption in the two states is given by {(CAH ,CAL), (CBH ,CBL)}, thenthe condition for equality between the marginal rates of substitution can bewritten as

U ′A(CAH )

U ′A(CAL)

= U′B(CBH )

U ′B(CBL)

.

The probabilities do not appear in this equation because, assuming A and Bhave the same probability beliefs, they appear as multipliers on both sides andso cancel out.

It is interesting to consider what happens in the case where one of theconsumers is risk neutral. We say that a consumer is risk neutral if his VNMutility function has the form

U (C) ≡ C .

A risk neutral consumer cares only about the expected value of his income orconsumption. In other words, his expected utility is just the expected valueof consumption. Suppose that consumer B is risk neutral. Then his marginalutility is identically equal to 1 in each state. Substituting this value into theefficiency condition, we see that

U ′A(CAH )

U ′A(CAL)

= 1,

which implies that CAH = CAL . All the risk is absorbed by the risk neutralconsumer B, leaving consumer A with a certain level of consumption.

Risk neutrality is a very special property, but there are circumstances inwhich risk averse consumers can achieve the same effects. First, let us con-sider more carefully the way in which the optimal consumption allocationdepends on income. First, note that the efficiency equation implies thatU ′A(CAH ) < U ′

A(CAL) if and only if U ′B(CBH ) < U ′

B(CBL). Since marginalutility is decreasing in consumption, this implies that CAH > CAL if and onlyif CBH > CBL . This immediately tells us that the optimal consumption alloca-tion depends only on the aggregate income of the pair. Let Ys = YAs + YBs for

2.2 Uncertainty 49

s = H , L. Then feasibility requires that total consumption equals total incomein each state:

CAs + CBs = Ys ,for s = H , L. Thus, the consumption of each consumer rises if and only ifaggregate income rises. This property is known as coinsurance between thetwo consumers: they provide insurance to each other in the sense that theirconsumption levels go up and down together. In particular, if the aggregateincome is the same in the two states, YH = YL , then CAH = CAL and CBH =CBL . So if aggregate income is constant, the consumption allocation will beconstant too, no matter how individual incomes fluctuate.

When there are only two consumers, constant aggregate income dependson a rather remarkable coincidence: when A’s income goes up, B’s incomegoes down by the same amount. When there is a large number of consumers,however, the same outcome occurs quite naturally, thanks to the law of largenumbers, as long as the incomes of the different consumers are assumed tobe independent. This is, in fact, what insurance companies do: they pool largenumbers of independent risks, so that the aggregate outcome becomes almostconstant, and then they can ensure that each individual gets a constant level ofconsumption. Suppose there is a large number of consumers i = 1, 2, ... withrandom incomes that are independently and identically distributed accordingto the probability distribution

Yi ={YH with probability πH

YL with probability πL .

Then the law of large numbers ensures that the average income is equal to theexpected value of an individual’s income Y = πHYH + πLYL with probabilityone, that is, with certainty. Then an insurance company could ensure every onea constant consumption level because the average aggregate income is almostconstant.

2.2.4 Portfolio choice

The use of Arrow securities to allocate income risk efficiently is a special caseof the portfolio choice problem that individuals have to solve in order todecide how to invest wealth in an uncertain environment. We can gain a lotof insight into the portfolio choice problem by considering the special case oftwo securities, one a safe asset and the other a risky asset.


As before, we assume there are two dates t = 0, 1 and two states s = H , Land a single consumption good at each date. Suppose that an investor has aninitial incomeW0 > 0 at date 0 and that he can invest it in two assets. One isa safe asset that yields one unit of the good at date 1 for each unit invested atdate 0. The other is a risky asset: one unit invested in the risky asset at date 0yields Rs > 0 units of the good in state s = H , L.We assume that the investor’srisk preferences are represented by a VNM utility function U (C) and that theprobability of state s is πs > 0 for s = H , L.

The investor’s portfolio can be represented by the fraction θ of his wealththat he invests in the risky asset. That is, his portfolio will contain θW0 unitsof the risky asset and (1 − θ)W0 of the safe asset. His future consumption willdepend on his portfolio choice and the realized return of the risky asset. LetCH and CL denote consumption in the high and low states, respectively. Then

Cs = RsθW0 + (1 − θ)W0,

for s = H , L. The investor chooses the portfolio that maximizes the expectedutility of his future consumption. That is, his decision problem is

maxθ πHU (CH ) + πLU (CL)

s.t. Cs = RsθW0 + (1 − θ)W0, s = H , L.

Substituting the expressions for CH and CL into the objective function we seethat the expected utility is a function of θ , say V (θ). The optimal portfolio0 < θ∗ < 1 satisfies the first-order condition V ′(θ∗) = 0, or

πHU′(CH )(RH − 1) + πLU

′(CL)(RL − 1) = 0.

The optimum is illustrated in Figure 2.10. The set of attainable consump-tion allocations (CH ,CL) is represented by the line segment with endpoints(W0,W0) and (RHW0,RLW0). If the investor puts all of his wealth in the safeasset θ = 0, then his future consumption in each state will beCH = CL = W0.If he puts all his wealth in the risky asset, then his future consumption with beRHW0 in the high state and RLW0 in the low state. If he puts a fraction θ ofhis wealth in the risky asset, his consumption bundle (CH ,CL) is just a convexcombination of these two endpoints with weights 1 − θ and θ respectively.In other words, we can trace out the line joining these two endpoints just byvarying the proportion of the risky asset between 0 and 1.

The optimal portfolio choice occurs where the investor’s indifference curveis tangent to the consumption curve. The tangency condition is just a geometricversion of the first-order condition above.

2.2 Uncertainty 51

CL

CH

(CH,CL)

0

Figure 2.10. Optimal choice of portfolio with two assets.

Depending on the investor’s risk preferences and the rates of return, theoptimal portfolio may consist entirely of the safe asset, entirely of the riskyasset, or a mixture of the two. It is interesting to see under which conditionseach of these possibilities arises. To investigate this question, we need to findout more about the slopes of the indifference curves and the feasible set.

The slope of the feasible set is easily calculated. Compare the portfolio inwhich all income is invested in the safe asset with the portfolio inwhich incomeis all invested in the risky asset. The change in CH is �CH = W0 −W0RH andthe change in CL is �CL = W0 −W0RL . So the slope is

�CL�CH

= W0 −W0RLW0 −W0RH

= 1 − RL1 − RH .

The slope is negative if RL < 1 < RH .An indifference curve is a set of points like (CH ,CL) that satisfy an

equation like

πHU (CH ) + πLU (CL) = constant.

Consider a“small”movement (dCH , dCL) along the indifference curve: it mustsatisfy the equation

πHU′(CH )dCH + πLU

′(CL)dCL = 0.

We can “solve” this equation to obtain the slope of the indifference curve:

dCLdCH

= −πHU ′(CH )

πLU ′(CL).


Now we are ready to characterize the different possibilities. For an interiorsolution, 0 < θ∗ < 1, the slope of the indifference curve must equal the slopeof the feasible set, or

πHU ′(CH )

πLU ′(CL)= 1 − RLRH − 1

.

A necessary and sufficient condition for this is that the indifference curve isflatter than the feasible set at (RHW0,RLW0) and steeper than the feasible setat (W0,W0). That is,

πHU ′(W0)

πLU ′(W0)>

1 − RLRH − 1

>πHU ′(RHW0)

πLU ′(RLW0).

The left hand inequality can be simplified to

πH

πL>

1 − RLRH − 1

or πHRH +πLRL > 1. In other words, the investor will hold a positive amountof the risky asset if and only if the expected return of the risky asset is greaterthan the return to the safe asset. This makes sense because there is no rewardfor bearing risk otherwise. The right hand inequality implies that

1 − RLRH − 1

> 0,

or RL < 1 < RH . In other words, the investor will hold the safe asset only ifthe risky asset produces a capital loss in the low state. Otherwise, the risky assetdominates (always pays a higher return than) the safe asset. Notice that evenif the risky asset sometimes yields a loss, the investor may choose to invest allhis wealth in the risky asset. It just depends on his attitude toward risk and therisk–return trade-off.

2.3 LIQUIDITY

The word liquidity is used in two senses here. First, we describe assets as liquidif they can be easily converted into consumption without loss of value. Second,we describe individuals as having a preference for liquidity if they are uncertainabout the timing of their consumption and hence desire to hold liquid assets.

2.3 Liquidity 53

Liquid assets

Once again, we illustrate the essential ideas using a simple example. Time isdivided into three periods or dates indexed by t = 0, 1, 2. At each date, there isa single all-purpose good which can be used for consumption or investment.

There are two assets that consumers can use to provide for future consump-tion, a short-term, liquid asset and a long-term, illiquid asset. In what follows,we refer to these as the short and long assets, respectively. Each asset is repre-sented by a constant-returns-to-scale investment technology. The short assetis represented by a storage technology that allows one unit of the good at datet to be converted into one unit of the good at date t + 1, for t = 0, 1. Thelong asset is represented by an investment technology that allows one unit ofthe good at date 0 to be converted into R > 1 units of the good at date 2. Weassume the return of the long asset is known with certainty. This assumptionsimplifies the analysis and allows us to focus attention on the other source ofuncertainty, that is, uncertainty about individual time preferences.

There is a trade-off between an asset’s time to maturity and its return. Thelong asset takes two periods to mature, but pays a high return. The short assetmatures after one period but yields a lower return. This trade-off is charac-teristic of the yield curve for bonds of different maturities, where we see thatbonds with short maturities typically have lower returns than bonds with longmaturities. The higher returns on the longer-dated assets can be interpretedboth as a reward for the inconvenience of holding illiquid assets and as areflection of the greater productivity of roundabout methods of production.

Liquidity preference

We model preference for liquidity as the result of uncertainty about timepreference. Imagine a consumer who has an endowment of one unit of thegood at date 0 and nothing at the future dates. All consumption takes place inthe future, at dates 1 and 2, but the consumer is uncertain about the precisedate at which he wants to consume. More precisely, we assume there are twotypes of consumers, early consumers who only want to consume at date 1 andlate consumers who only want to consume at date 2. Initially, the consumerdoes not know his type. He only knows the probability of being an early or alate consumer. Let λ denote the probability of being an early consumer and1 − λ the probability of being a late consumer. The consumer learns whetherhe is an early or late consumer at the beginning of date 1.

Uncertainty about time preferences is a simple way of modeling whateconomists call a “liquidity shock,” that is, an unanticipated need for liquid-ity resulting from an event that changes one’s preferences. This could be an


accident that requires an immediate expenditure, the arrival of an unexpectedinvestment opportunity, or an unexpected increase in the cost of an expend-iture that was previously planned. We can think of λ as measuring the degreeof a consumer’s liquidity preference. Other things being equal, he will wantto earn the highest return possible on his investments. But if he is uncertainabout the timing of his consumption, we will also care about liquidity, thepossibility of realizing the value of this asset at short notice. If λ is one, theconsumer’s liquidity preference will be high, since he cannot wait until date 2to earn the higher return on the long asset. If λ is zero, he will have no pref-erence for liquidity, since he can hold the long asset without inconvenience.For λ between zero and one, the consumer’s uncertainty about the timing ofhis consumption poses a problem. If the consumer knew that he was a lateconsumer, he would invest in the long asset because it gives a higher return.If he knew that he was an early consumer, he would hold only the short assetin spite of its lower return. Since the consumer is uncertain about his type, hewill regret holding the short asset if he turns out to be a late consumer andhe will regret holding the long asset if he turns out to be an early consumer.The optimal portfolio for the consumer to hold will depend on both his riskaversion and his liquidity preference and on the return to the long asset (theslope of the yield curve).

Investment under autarky

Suppose the consumer has a period utility function U (C) and let C1 and C2

denote his consumption at date 1 (if he is an early consumer) and date 2 (ifhe is a late consumer). Then his expected utility from the consumption stream(C1,C2) is

λU (C1) + (1 − λ)U (C2).

His consumption at each date will be determined by his portfolio choice atdate 0. Let θ denote the proportion of his wealth invested in the short asset.Recall that he has an initial endowment of one unit of the good at date 0 so heinvests θ in the short asset and 1 − θ in the long asset. Then his consumptionat date 1 is given by

C1 = θ ,

since he cannot consume the returns to the long asset,whereas his consumptionat date 2 is given by

C2 = θ + (1 − θ)R,

2.3 Liquidity 55

since the returns to the short asset can be re-invested at date 1 and consumedat date 2. Note that C1 < C2 except in the case where θ = 1, so the consumerfaces some risk and if he is risk averse this will impose some loss of expectedutility compared to a situation in which he can consume the expected valueC = λC1 + (1 − λ)C2 for sure. Of course, he can choose θ = 1 if he wantsto avoid uncertainty altogether, but there is a cost to doing so: his averageconsumption will be lower.

The consumer’s decision problem is to choose θ to maximize

λU (θ) + (1 − λ)U (θ + (1 − θ)R).

At an interior solution, the optimal value of θ satisfies

λU ′(θ) + (1 − λ)U ′(θ + (1 − θ)R) (1 − R) = 0.

For example, if U (C) = lnC , then the first-order condition becomes

λ

θ+ (1 − λ)

θ + (1 − θ)R(1 − R) = 0,

or

θ = λR

R − 1.

So the fraction of wealth held in the short asset is greater if λ is greater andlower if R is greater. Note that θ = 0 for any value of λ greater than 1 − 1/R.

Risk pooling

As we have seen already, the consumer’s attempt to provide for his futureconsumption needs is bound to lead to regret as long as he cannot perfectlyforesee his type. If he could ensure against his liquidity shock, he could dobetter. Suppose that there is a large number of consumers, all of whom are exante identical and subject to the same shock, that is, they all have a probabilityλ of being early consumers. If we assume further that their liquidity shocksare independent, then the law of large numbers assures us that there will beno aggregate uncertainty. Whatever happens to the individual consumer, thefraction of the total population who become early consumers will be λ forcertain. This suggests the potential for pooling risks and providing a bettercombination of returns and liquidity.

To see how this would work, suppose a financial institution were to takecharge of the problem of investing the endowments of a large number of


consumers and providing for their consumption. The financial institutionwould take the endowments at date 0 and invest a fraction θ in the short assetand a fraction 1 − θ in the long asset. At date 1 it would provide consumptionequal toC1 units of the good to early consumers and at date 2 it would provideC2 units of the good to late consumers. The important difference betweenthe financial institution and the individual consumers is that the companyfaces no uncertainty: it knows for sure that a fraction λ of its clients willbe early consumers. Consequently, it knows for sure what the demand forthe consumption good will be at date 1 and date 2. At date 1 it needs toprovide λC1 per capita and at date 2 it needs to provide (1 − λ)C2 per capita.Because the return to the short asset is lower than the return to the long asset,the financial institution will hold the minimum amount of the short assetit needs to provide for the early consumers’ consumption at date 1, that is,θ = λC1 and it will hold the rest of the portfolio in the long asset. Then thecompany’s plans are feasible if

λC1 = θ

and

(1 − λ)C2 = (1 − θ)R.

The company’s decision problem is

max λU (C1) + (1 − λ)U (C2)

s.t. λC1 = θ

(1 − λ)C2 = (1 − θ)R.

If we substitute for the C1 and C2 in the objective function, we get

λU

(θ

λ

)+ (1 − λ)U

((1 − θ)R

1 − λ

)

and the first-order condition formaximizing this expressionwith respect to θ is

U ′ (C1) − U ′ (C2)R = 0.

Notice that the terms involving λ have cancelled out. In terms of the earlierexample, where the period utility function is U (C) = lnC , the first-ordercondition implies that C2 = C1R, or

θ = λ.



This chapter has provided a foundation in terms of the basic finance andeconomics that will be needed to understand this book. Those wishing topursue these topics further can consult a textbook such as Mas-Collel et al.(1995).

REFERENCE

Mas-Collel, A., M. Whinston, and J. Green (1995). Microeconomic Theory, Oxford:Oxford University Press.

3

Intermediation and crises

With the exception of the Bretton Woods period from 1945 to 1971, bank-ing crises have occurred fairly frequently in the past 150 years, as we saw inChapter 1. There are two traditional ways of understanding crises. The firstasserts that crises result from panics; the second asserts that crises arise fromfundamental causes that are part of the business cycle. Both have a long history.For example, Friedman and Schwartz (1963) and Kindleberger (1978) arguedthat many banking crises resulted from unwarranted panics and that most ofthe banks that were forced to close in such episodes were illiquid rather thaninsolvent. The alternative view, put byMitchell (1941) and others, is that finan-cial crises occur when depositors have reason to believe that economic funda-mentals in the near future look poor. In that case depositors, anticipating thatfuture loan defaults will make it impossible for the bank to repay their deposits,withdraw their money now. The depositors in this case are anticipatinginsolvency rather than illiquidity. In this chapter,we consider both approaches.

Although the economic theory of banking goes back over 200 years, it wasnot until recently that a model of banking, in the contemporary sense, wasprovided in the seminal papers of Bryant (1980) and Diamond and Dybvig(1983). The publication of these papers marked an important advance in thetheory of banking. Although the objective of the papers was to provide anexplanation of bank runs, an equally important contribution was to providea microeconomic account of banking activity that was distinct from otherfinancial institutions. In fact, the papers contributed four separate elements tothe theory of banking:

• a maturity structure of bank assets, in which less liquid assets earn higherreturns;

• a theory of liquidity preference, modeled as uncertainty about the timing ofconsumption;

• the representation of a bank as an intermediary that provides insurance todepositors against liquidity (preference) shocks;

• an explanation of bank runs by depositors. In the case of Diamond andDybvig (1983), the bank runs are modeled as the result of self-fulfillingprophecies or panics; in the case of Bryant (1980), they are modeled as theresult of fundamentals.

3.1 The Liquidity Problem 59

In Sections 3.1–3.4 of this chapter, we describe a model of banking, looselybased on the Bryant (1980) and Diamond–Dybvig (1983) models, that showshowall these pieces fit together (seeChapter 2 for an introductory developmentof some of these ideas). Sections 3.5 and 3.6 develop amodel based on the viewthat crises arise from panics while Section 3.7 develops a model based on theview that crises result from fundamentals. Section 3.8 contains a literaturereview and Section 3.9 concluding remarks.

3.1 THE LIQUIDITY PROBLEM

It is a truism that banks have liquid liabilities and illiquid assets. In otherwords, they borrow short and lend long. This makes banks vulnerable to sud-den demands for liquidity (bank runs), but more on this later. This maturitymismatch reflects the underlying structure of the economy: individuals have apreference for liquidity but the most profitable investment opportunities takea long time to pay off. Banks are an efficient way of bridging the gap betweenthe maturity structure embedded in the technology and liquidity preference.

We adopt the period structure introduced inChapter 2. There are three datesindexed by t = 0, 1, 2. At each date there is a single good that can be used forconsumption and investment and serves as a numeraire. There are two typesof assets:

• The liquid asset (also called the short asset ) is a constant returns to scaletechnology that takes one unit of the good at date t and converts it into oneunit of the good at date t + 1, where t = 0, 1.

• The illiquid asset (also called the long asset ) is a constant returns to scaletechnology that takes one unit of the good at date 0 and transforms it intoR > 1 units of the good at date 2; if the long asset is liquidated prematurelyat date 1 then it pays 0 < r < 1 units of the good for each unit invested.

At the first date there is a large number, strictly speaking a continuum, of exante identical economic agents (consumers, depositors).1 Each consumer hasan endowment of one unit of the good at date 0 and nothing at the later dates.In order to provide for future consumption, agents will have to invest, directlyor indirectly, in the long and short assets.

1 We represent the set of agents by the unit interval [0, 1], where each point in the intervalis a different agent. We normalize the measure of the entire set of agents to be equal to oneand measure the fraction of agents in any subset by its Lebesgue measure. The assumption of alarge number of individually insignificant agents ensures perfect competition, that is, no one hasenough market power to affect the equilibrium terms of trade.

60 Chapter 3. Intermediation and Crises

The agents’ time preferences are subject to a random shock at the beginningof date 1. With probability λ an agent is an early consumer, who only valuesconsumption at date 1; with probability (1−λ) he is a late consumer who onlyvalues consumption at date 2. The agent’s (random) utility function u(c1, c2)is defined by

u(c1, c2) ={U (c1) w.pr. λU (c2) w.pr. 1 − λ,

where ct ≥ 0 denotes consumption at date t = 1, 2 and U (·) is a neoclassicalutility function (increasing, strictly concave, twice continuously differentiable).Because there is a large number of agents and their preference shocks areindependent, the ‘law of large numbers’ holds and we assume that the fractionof early consumers is constant and equal to the probability of being an earlyconsumer. Then, although each agent is uncertain about his type, early or late,there is no uncertainty about the proportion of each type in the population.With probability one there is a fraction λ of early consumers and a fraction1 − λ of late consumers.

This simple example illustrates the problem of liquidity preference. If anagent knew his type at date 0, he could invest in the asset that would providehim with consumption exactly when he needed it. For example, if he were anearly consumer, he could invest his endowment in the short asset and it wouldproduce a return of one unit at date 1, exactly when he wanted to consume it.If he were a late consumer, he could invest his endowment in the long asset,which yields a higher rate of return, and it would produce R units at date 2exactly when he wanted to consume it. The problem is precisely that he doesnot know his preferences regarding the timing of consumption when he hasto make the investment decision. If he invests in the short asset and turns outto be a late consumer, he will regret not having invested in the higher-yieldinglong asset. If he invests in the long asset and turns out to be an early consumer,he will have nothing to consume and will clearly regret not having invested inthe short asset. Even if he invests in a mixture of the two assets, he will stillhave some regrets with positive probability. It is this problem – the mismatchbetween asset maturity and time preferences – that financial intermediation isdesigned to solve.

3.2 MARKET EQUILIBRIUM

Before we consider the use of intermediaries to solve the liquidity problem,we consider a market solution. When we laid out the problem of matchinginvestment maturities and time preferences in Chapter 2, we assumed that the

3.2 Market Equilibrium 61

agent existed in a state of autarky, that is, he was unable to trade assets andhad to consume the returns generated by his own portfolio. The assumptionof autarky is unrealistic. An agent often has the option of selling assets in orderto realize their value in a liquid form. In fact, one of the main purposes of assetmarkets is to provide liquidity to agents whomay be holding otherwise illiquidassets. So it is interesting to consider what would happen if long-term assetscould be sold (liquidated) on markets and see whether this solves the problemof matching maturities to time preferences.

In this section, we assume that there exists a market on which an agent cansell his holding of the long asset at date 1 after he discovers his true type. Then,if he discovers that he is an early consumer, he can sell his holding of the longasset at the prevailing price and consume the proceeds. The existence of anasset market transforms the illiquid long asset into a liquid asset, in the sensethat it can be sold for a predictable and sure price if necessary.

The possibility of selling the long asset at date 1 provides some insuranceagainst liquidity shocks. Certainly, the agent must be at least as well off as heis in autarky and he may be better off. However, the asset market at date 1cannot do as well as a complete set of contingent claims markets. For example,an agent might want to insure himself by trading goods for delivery at date 1contingent on the event that he is an early or late consumer. As we shall see,if an agent can trade this sort of contingent claims at date 0, before his type isknown, he can synthesize an optimal risk contract. The absence of markets forcontingent claims in the present set up explains the agent’s failure to achievethe first best.

At date 0, an investor has an endowment of one unit of the goodwhich can beinvested in the short asset or the long asset to provide for future consumption.Suppose he invests in a portfolio (x , y) consisting of x units of the long assetand y units of the short asset. His budget constraint at date 0 is

x + y ≤ 1.

At date 1 he discovers whether he is an early or late consumer. If he is an earlyconsumer, he will liquidate his portfolio and consume the proceeds. The priceof the long asset is denoted by P . The agent’s holding of the short asset provideshim with y units of the good and his holding of the long asset can be sold forPx units of the good. Then his consumption at date 1 is given by the budgetconstraint

c1 = y + Px .If the agent is a late consumer, he will want to rebalance his portfolio. Incalculating the optimal consumption for a late consumer, there is no essentialloss of generality in assuming that he always chooses to invest all of his wealth


in the long asset at date 1. This is because the return on the short asset (betweendates 1 and 2) is weakly dominated by the return on the long asset. To see this,note that if, contrary to our claim, the return on the short asset were greaterthan the return on the long asset, no one would be willing to hold the long assetat date 1 and the asset market could not clear. Thus, in equilibrium, either thetwo rates of return are equal or the long asset dominates the short asset (andno one holds the short asset at date 1). This ordering of rates of return impliesthat P ≤ R in equilibrium and that it is weakly optimal for the late consumerto hold only the long asset at date 1. Then the quantity of the long asset heldby the late consumer is x + y/P since he initially held x units of the long assetand he exchanges the return on the short asset for y/P units of the long asset.His consumption at date 2 will be given by the budget constraint

c2 =(x + y

P

)R.

From the point of view of an investor at date 0, the objective is to choose aportfolio (x , y) to maximize the investor’s expected utility

λU(y + Px)+ (1 − λ)U

[(x + y

P

)R].

The investor’s decision at date 0 canbe simplified if wenote that, in equilibrium,the price of the long assetmust be P = 1. To see this, note that if P > 1 then thelong asset dominates the short asset and no one will hold the short asset atdate 0. In that case, early consumers will be offering the long asset for salebut there will be no buyers at date 1. Then the price must fall to P = 0, acontradiction. On the other hand, if P < 1, then the short asset dominates thelong asset and no one will hold the long asset at date 0. Early consumers willconsume the returns on the short asset at date 1, realizing a return of 1 > P ;late consumers will try to buy the long asset to earn a return of R/P > R. Sinceno one has any of the long asset, the price will be bid up to P = R, anothercontradiction. Thus,P = 1 in equilibrium.At this price, the two assets have thesame returns and are perfect substitutes. The agent’s portfolio choice becomesimmaterial. The agent’s consumption is

c1 = x + Py = x + y = 1

at date 1 and

c2 =(x + y

P

)R = (x + y)R = R

3.2 Market Equilibrium 63

at date 2. Then the equilibrium expected utility is

λU (1) + (1 − λ)U (R).

This level of welfare serves as a benchmark against which to measure thevalue of having a banking system that can provide insurance against liquidity(preference) shocks.

The value of the market

It is obvious that the investor is at least as well off when he has access to theasset market as he is in autarky. Typically, he will be better off. To show this,we have to compare the market equilibrium allocation (c1, c2) = (1,R) withthe set of allocations that are feasible in autarky.

Aswe saw inChapter 2, if the consumer does not have access to the assetmar-ket and is forced to remain in autarky, his consumption as an early consumerwould be equal to his investment y in the safe asset, and his consumption as alate consumer would be equal to the return Rx = R(1 − y) on his investmentin the long asset plus his investment in the safe asset y which can be re-investedat the second date. Thus, the feasible consumption bundles have the form

c1 = yc2 = y + R(1 − y)

for some feasible value of y between 0 and 1. The set of such consumptionbundles is illustrated in Figure 3.1.

c2

c10 1

1

R

Market allocation

Feasible set under autarky

Figure 3.1. Comparison of the market allocation with the feasible set under autarky.


As the figure illustrates, themaximumvalue of early consumption is attainedwhen y = 1 and c1 = 1 and themaximumvalue of late consumption is attainedwhen y = 0 and c2 = R. The market allocation (c1, c2) = (1,R) dominatesevery feasible autarkic allocation, that is, it gives strictly greater consumptionat one date and typically gives greater consumption at both. Thus, access tothe asset market does increase expected utility.

3.3 THE EFFICIENT SOLUTION

The market provides liquidity by allowing the investor to convert his holdingof the long asset into consumption at the price P = 1 at the second date.Because the asset market is perfectly competitive, the investor can buy and sellany amount of the asset at the equilibrium price. This means that the marketis perfectly liquid in the sense that the price is insensitive to the quantity ofthe asset that is traded. However, it turns out that the provision of liquidity isinefficient. We will discuss the reasons for this in greater detail later, but theshort explanation for this inefficiency is that the set of markets in the economydescribed above is incomplete. In particular, there is no market at date 0 inwhich an investor can purchase the good for delivery at date 1 contingent onhis type. If such a market existed, the equilibrium would be quite different.

We take as our definition of the efficient provision of liquidity, the level ofwelfare that could be achieved by a central planner who had command of allthe allocation decisions in the economy. To begin with, we assume that theplanner has complete information about the economy, including the abilityto tell who is an early consumer and who is a late consumer. This importantassumption will later be relaxed.

The central planner chooses the amount per capita x invested in the longasset and the amount per capita y invested in the short asset. Then he choosesthe consumption per capita c1 of the early consumers at date 1 and the con-sumption per capita c2 of the late consumers at date 2. The central planneris not bound to satisfy any equilibrium conditions. He is only constrained bythe condition that the allocation he chooses must be feasible. At date 0 thefeasibility condition is simply that the total invested per capita must be equalto the per capita endowment:

x + y = 1. (3.1)

At the second date, the feasibility condition is that the total consumption percapita must be less than or equal to the return on the short asset. Since thefraction of early consumers is λ and each one is promised c1 the consumption

3.3 The Efficient Solution 65

per capita (i.e. per head of the entire population) is λc1. Then the feasibilitycondition is

λc1 ≤ y . (3.2)

If this inequality is strict, some of the good can be re-invested in the shortasset and consumed at date 2. Thus, the total available at date 2 (expressedin per capita terms) is Rx + (y − λc1). The total consumption per capita(i.e. per head of the entire population) at date 2 is (1 − λ)c2 since the fractionof late consumers is 1 − λ and each of them is promised c2. So the feasibilitycondition is

(1 − λ)c2 ≤ Rx + (y − λc1),

which can also be written as

λc1 + (1 − λ)c2 ≤ Rx + y . (3.3)

The planner’s objective is to choose the investment portfolio (x , y) and theconsumption allocation (c1, c2) to maximize the typical investor’s expectedutility

λU (c1) + (1 − λ)U (c2),

subject to the various feasibility conditions (3.1)–(3.3).This looks like a moderately complicated problem, but it can be simplified

if we use a little common sense. The first thing to note is that it will never beoptimal to carry over any of the short asset from date 1 to date 2. To see this,suppose that y > λc1 so that some of the good is left over at date 1. Then wecould reduce the amount invested in the short asset at date 0 by ε > 0 say andinvest it in the long asset instead. At date 2, there would be ε less of the shortasset but ε more of the long asset. The net change in the amount of goodsavailable would be Rε − ε = (R − 1)ε > 0, so it would be possible to increasethe consumption of the late consumers without affecting the consumption ofthe early consumers. This cannot happen in an optimal plan, so it follows thatin any optimal plan we must have λc1 = y and (1 − λ)c2 = Rx . Thus, once xand y are determined, the optimal consumption allocation is also determinedby the relations

c1 = y

λ;

c2 = Rx

1 − λ.


(Recall that y is the return on the short asset per head of the entire population,whereas c1 is the consumption of a typical early consumer, so c1 is greater thany . Similarly, the consumption c2 of a typical late consumer is greater than thereturn on the long asset per head of the entire population.) If we substitute theseexpressions for consumption into the objective function and use the date-0feasibility condition to write x = 1 − y , we can see that the planner’s problemboils down to choosing y in the interval [0, 1] to maximize

λU( yλ

)+ (1 − λ)U

(R(1 − y)1 − λ

). (3.4)

Ignoring the possibility of a boundary solution, where y = 0 or y = 1,a necessary condition for an optimal choice of y is that the derivative of thefunction in (3.4) be zero.Differentiating this function and setting the derivativeequal to zero yields

U ′ ( yλ

)− U ′

(R(1 − y)1 − λ

)R = 0,

or, substituting in the consumption levels,

U ′ (c1) = U ′ (c2)R. (3.5)

There are several interesting observations to be made about this first-ordercondition. First, the value of λ does not appear in this equation: λ drops outwhen we differentiate the objective function. The intuition for this result isthat λ appears symmetrically in the objective function and in the feasibilityconditions. An increase in λ means that early consumers get more weight inthe objective function but it also means that they cost more per capita tofeed. These two effects cancel out and leave the optimal level of consumptionunchanged.

The inefficiency of the market solution

The second point to note concerns the (in)efficiency of the market solution.The set of feasible consumption allocations for the planner’s problem is illus-trated in Figure 3.2. For each choice of y in the interval between 0 and 1, theconsumption allocation is defined by the equations

c1 = y

λ;

c2 = R(1 − y)(1 − λ)

.


c2

c10 1

1

R

R/(1 – λ)

1/λ

Market allocation

Efficient allocation given these preferences

Figure 3.2. Comparison of the market allocation with the efficient allocation.

The consumption of the early consumers is maximized by putting y = 1, inwhich case (c1, c2) = (1/λ, 0). Similarly, the consumption of the late con-sumers is maximized by putting y = 0, in which case (c1, c2) = (0,R/(1−λ)).Since the equations for consumption are linear in y , we can attain any pointon the line segment joining (1/λ, 0) and (0,R/(1 − λ)). This feasible frontieris described in the figure. The efficient point is determined by the tangency ofthe consumers’ indifference curve with the feasible frontier. Depending on theconsumers’ preferences, the point of tangency could occur anywhere along thefeasible frontier.

The market allocation occurs on the feasible frontier. Simply put y = λ andwe get

(c1, c2) =(y

λ,R(1 − y)(1 − λ)

)= (1,R) .

This allocation could be efficient but typically it will not be. To see this, supposethat by some chance the market equilibrium resulted in the same allocation asthe planner’s problem. Then the first-order condition (3.5) becomes

U ′(1) = U ′(R)R.


In some special cases this conditionmay be satisfied. For example, suppose thatthe investor has a logarithmic utility function U (c) = ln c so that U ′(c) =1/c . Substituting this expression in the preceding equation, the left hand sidebecomesU ′(1) = 1 and the right hand side becomesU ′(R)R = (1/R)R = 1.In this particular case, the market provision of liquidity is efficient: a centralplanner could do no better than the market. For other utility functions, thiswould not be the case. Suppose, for example, that the investor’s utility functionbelongs to the constant relative risk aversion class

U (c) = 1

1 − σc1−σ

where σ > 0 is the degree of relative risk aversion. Then U ′(c) = c−σ

and substituting this into the necessary condition for efficiency, we see thatthe left hand side becomes U ′(1) = 1 and the right hand side becomesU ′(R)R = R−σR = R1−σ . Except in the case σ = 1 (which correspondsto the logarithmic case), R1−σ �= 1 and the allocation chosen by the plannermust be different from themarket allocation. So for any degree of risk aversiondifferent from 1 the planner achieves a strictly better level of expected utilitythan the market.

Liquidity insurance

A third insight that can be derived from the first-order condition (3.5) concernsthe provision of insurance against liquidity shocks, that is, the event of being anearly consumer. Even in the efficient allocation the individual faces uncertaintyabout consumption. The first-order condition U ′(c1) = RU ′(c2) implies thatc1 < c2 so the individual’s consumption will be higher or lower depending onwhether he is a late or an early consumer.

In the market allocation, the early consumers get 1 and the late consumersget R > 1 so it is clearly worse to be an early consumer than a late consumer. Arisk averse investor would like to havemore consumption as an early consumerand less as a late consumer, assuming that the expected value of his consumptionremains the same. An interesting question is whether the planner providesinsurance against the liquidity shock by reducing the volatility of consumption,that is, increasing consumption of early consumers and reducing consumptionof late consumers relative to themarket solution. In Figure 3.2, the consumptionallocations that lie to the right of the market allocation all satisfy c1 > 1 andc2 < R, that is, they have less uncertainty about consumption than the marketallocation. On the other hand, they also have lower expected consumption.

It is interesting to see under what conditions the optimal deposit contractgives the consumer more consumption at date 1 than the market solution, thatis,when are the higher returns from the long asset shared between the early and


late consumers. Substituting from the budget constraints into the first-ordercondition we get the equation

U ′ ( yλ

)= RU ′

((1 − y)(1 − λ)

R

).

If y = λ this condition reduces to U ′(1) = RU ′(R), so a necessary andsufficient condition for c1 = y/λ > 1 and c2 = R(1 − y)/(1 − λ) < R is that

U ′(1) > RU ′(R).

A sufficient condition is that cU ′(c) be decreasing in c , which is equivalent tosaying that the relative risk aversion is greater than one:

η(c) ≡ − cU′′(c)

U ′(c)> 1.

If this inequality is reversed and η(c) < 1, early consumers get less and lateconsumers get more than in the benchmark case, that is,

c1 = y/λ < 1,

c2 = R(1 − y)/(1 − λ) > R.

These results have an intuitive explanation. The logarithmic utility function,which has a degree of relative risk aversion equal to unity,marks the boundarybetween two different cases. If relative risk aversion equals one, as in the nat-ural log case, the market outcome is efficient and, in particular, the market’sprovision of liquidity is optimal. If relative risk aversion is greater than one,the market provision of liquidity is inefficient. An efficient allocation shouldprovidemore insurance by increasing the consumption of the early consumersand reducing the consumption of the late consumers. If relative risk aversionis less than one, there is paradoxically too much liquidity in the sense thatefficiency requires a reduction in the consumption of the early consumers andan increase in the consumption of the late consumers.

This last result alerts us to the fact that insurance is costly. In order to providemore consumption to the early consumers, it is necessary to hold more of theshort asset and, hence, less of the long asset. Since the long asset’s return isgreater than the short asset’s, the increase in the amount of the short asset inthe planner’s portfolio must reduce average consumption across the two dates.As long as relative risk aversion is greater than one, the benefit from insuranceis greater than the cost, at least to start with. If the relative risk aversion is lessthan one, the benefits of greater insurance are not worth the cost and, indeed,


the efficient allocation requires the planner to increase the risk borne by theinvestors in order to capture the increased returns from holding the long asset.

Why do the investors hold the wrong portfolio in themarket solution? Theirportfolio decision is dependent upon themarket price for the long asset P = 1.This price is determinedby the condition that investors at date 0must bewillingto hold both assets. The market does not reveal how much investors would bewilling to pay for the asset contingent on knowing their type. Consequently,the price does not reflect the value of being able to sell the long asset as an earlyconsumer or being able to buy it as a late consumer.

Complete markets

We mentioned earlier (in Section 3.2) the possibility of introducing marketsthat would allow individual agents to trade at date 0 claims on date-1 con-sumption contingent on their type, early or late. The existence of such marketswould achieve the same allocation of risk and the same portfolio investmentas the central planner. Unlike the model economy in which there are no con-tingent markets, an economy with markets for individual contingencies wouldsignal the correct value of each asset to the market, in particular the valueof liquidity, and ensure that the efficient allocation was achieved. To see this,suppose that an individual can purchase date-1 consumption at a price q1 if heis early and q2 if he is late. Note that these prices are measured in terms of thegood at date 0. The implicit price of goods at date 2 in terms of goods at date 1is p = P/R, as usual. Then the budget constraint for an individual at date 0 is

q1λC1 + q2p (1 − λ)C2 ≤ 1. (3.6)

The left hand side represents the present value (at date 0) of expected con-sumption (since there is no aggregate uncertainty, each individual only paysfor the expected value of his demand for goods at each date). With probabil-ity λ he demands C1 units of date-1 consumption and the present value ofλC1 is q1λC1. Similarly, with probability 1 − λ he demands C2 units of date-2consumption and the present value of (1 − λ)C2 is q2p (1 − λ)C2.

The individual chooses (C1,C2) to maximize λU (C1) + (1 − λ)U (C2)

subject to (3.6) and the solution must satisfy the first-order conditions

λU ′ (C1) = µq1λ

and

(1 − λ)U ′ (C2) = µq2p (1 − λ)


where µ > 0 is the Lagrange multiplier associated with the constraint. Then

U ′ (C1)

U ′ (C2)= q1q2p

.

Since the investment technology exhibits constant returns to scale, the equi-librium prices must satisfy two no-arbitrage conditions. To provide one unitof the good at date 1, it is necessary to invest 1 unit in the short asset at date0. Thus, there are zero profits from investing in the short asset if and only ifq1 = 1. Similarly, to provide one unit of the good at date 2, it is necessary toinvest 1/R units in the long asset at date 0. Thus, there are zero profits frominvesting in the long asset if and only if pq2 = 1/R. This implies that

U ′ (C1)

U ′ (C2)= R,

the condition required for efficient risk sharing.

Private information and incentive compatibility

So far we have assumed that the central planner knows everything, includingwhether an investor is an early or late consumer. This allows the planner toassign different levels of consumption to early and late consumers. Since aninvestor’s time preferences are likely to be private information, the assumptionthat the planner knows the investor’s type is restrictive. If we relax this assump-tion, we run into a problem: if time preferences are private information, howcan the planner find out who is an early consumer and who is a late consumer?The planner can rely on the individual truthfully revealing his type if and onlyif the individual has no incentive to lie. In other words, the allocation chosenby the planner must be incentive-compatible.

In the present case, it is quite easy to show that the optimal allocationis incentive compatible. First, the early consumers have no opportunity tomisrepresent themselves as late consumers. A late consumer is given c2 atdate 2 and since the early consumer only values consumption at date 1 hewould certainly be worse off if he waited until date 2 to receive c2. The lateconsumer poses more of a problem. He could pretend to be an early consumer,receive c1 at date 1 and store it until date 2 using the short asset. To avoidgiving the late consumer an incentive to misrepresent his preferences, he mustreceive at least as much consumption as the early consumer. This means thatthe allocation is incentive compatible if and only if

c1 ≤ c2. (3.7)


Fortunately for us, if we consult the first-order condition for the planner’sproblem, equation (3.5), we find that it implies that c1 < c2 since R > 1and U ′′(c) < 0 so that the optimal allocation is automatically incentive-compatible.

The allocation that optimizes the typical investor’s welfare subject to theincentive constraint (3.7) is called incentive-efficient. In this case, becausethe incentive constraint is not binding at the optimum, the incentive-efficientallocation is the same as the first-best allocation.

Although the incentive-compatibility condition does not have any effect onthe optimal risk-sharing allocation, private information plays an importantrole in the account of banking that we give in the sequel. In particular, thefact that a bank cannot distinguish early and late consumers means that allconsumers can withdraw from the bank at date 1 and this is a crucial featureof the model of bank runs.

3.4 THE BANKING SOLUTION

A bank, by pooling the depositors’ investments, can provide insurance againstthe preference shock and allow early consumers to share the higher returns ofthe long asset. The bank takes one unit of the good from each agent at date 0and invests it in a portfolio (x , y) consisting of x units of the long asset and yunits of the short asset. Because there is no aggregate uncertainty, the bank canoffer each consumer a non-stochastic consumption profile (c1, c2), where c1 isthe consumption of an early consumer and c2 is the consumption of a late con-sumer.We can interpret (c1, c2) as a deposit contract underwhich the depositorhas the right to withdraw either c1 at date 1 or c2 at date 2, but not both.

There is assumed to be free entry into the banking sector. Competitionamong the banks forces them to maximize the ex ante expected utility ofthe typical depositor subject to a zero-profit (feasibility) constraint. In fact,the bank is in exactly the same position as the central planner discussedin the previous section. At date 0 the bank faces a budget constraint

x + y ≤ 1. (3.8)

At date 1, the bank faces a budget constraint

λc1 ≤ y . (3.9)

Recalling that it is never optimal to carry consumption over from date 1 todate 2 by holding the short asset, we can write the budget constraint for the

3.4 The Banking Solution 73

bank at the third date as

(1 − λ) c2 ≤ Rx . (3.10)

Formally, the bank’s problem is to maximize the expected utility of the typicaldepositor

λU (c1) + (1 − λ)U (c2)

subject to the budget constraints (3.8)–(3.10).We do not explicitly impose the incentive-compatibility constraint because,

as we saw previously, the solution to the unconstrained optimization problemwill automatically satisfy the incentive constraint

c1 ≤ c2.

So the bank is able to achieve the first-best allocation on behalf of its customers.It is worth pausing to note how this account of bank behavior implements

three of the four elements of banking theory mentioned at the beginning ofthis chapter.

• It provides a model of the maturity structure of bank assets, in which lessliquid assets earn higher returns. In this case, there are two bank assets, theliquid short asset, which yields a return of 1, and the illiquid long asset,which yields a return of R > 1.

• It provides a theory of liquidity preference,modeled as uncertainty about thetiming of consumption. Thematuritymismatch arises because an investor isuncertain of his preferences over the timing of consumption at the momentwhen an investment decision has to be made.

• It represents the bank as an intermediary that provides insurance to depos-itors against liquidity (preference) shocks. By pooling his resources withthe bank’s and accepting an insurance contract in the form of promises ofconsumption contingent on the date of withdrawal, the investor is able toachieve a better combination of liquidity services and returns on investmentthan he could achieve in autarky or in the asset market.

The properties of the efficient allocation, derived in the preceding section,of course apply to the banking allocation, so we will not repeat them here.Instead, we want to focus on the peculiar fragility of the arrangement that thebank has instituted in order to achieve optimal risk sharing.


3.5 BANK RUNS

At the beginning of this chapter, we mentioned four contributions to bankingtheorymade by the seminal papers of Bryant (1980) andDiamond andDybvig(1983).Wehave discussed the first three and nowwe turn to the fourth,namely,the explanation of bank runs. In this section, we develop a model of bank runsas panics or self-fulfilling prophecies. Later we shall consider bank runs as theresult of fundamental forces arising in the course of the business cycle.

Suppose that (c1, c2) is the optimal deposit contract and (x , y) is the optimalportfolio for the bank. In the absence of aggregate uncertainty, the portfolio(x , y) provides just the right amount of liquidity at each date assuming thatthe early consumers are the only ones to withdraw at date 1 and the lateconsumers all withdraw at date 2. This is an equilibrium in the sense thatthe bank is maximizing its objective, the welfare of the typical depositor, andthe early and late consumers are timing their withdrawals to maximize theirconsumption.

So far, we have treated the long asset as completely illiquid: there is no waythat it can be converted into consumption at date 1. Suppose, instead, thatthere exists a liquidation technology that allows the long-term investment tobe terminated prematurely at date 1. More precisely, we assume that

• if the long asset is liquidated prematurely at date 1, one unit of the long assetyields r ≤ 1 units of the good.

Under the assumption that the long asset can be prematurely liquidated, witha loss of R− r per unit, there exists another equilibrium if we also assume thatthe bank is required to liquidate whatever assets it has in order to meet thedemands of the consumers who withdraw at date 1. To see this, suppose thatall depositors, whether they are early or late consumers, decide to withdraw atdate 1. The liquidated value of the bank’s assets at date 1 is

rx + y ≤ x + y = 1

so the bank cannot possibly pay all of its depositors more than one unit atdate 1. In the event that c1 > rx + y , the bank is insolvent and will be able topay only a fraction of the promised amount. More importantly, all the bank’sassets will be used up at date 1 in the attempt to meet the demands of theearly withdrawers. Anyone who waits until the last period will get nothing.Thus, given that a late consumer thinks everyone else will withdraw at date 1it is optimal for a late consumer to withdraw at date 1 and save the proceedsuntil date 2. Thus, bank runs are an equilibrium phenomenon. The followingpayoff matrix illustrates the two equilibria of this coordination game. The rows

3.5 Bank Runs 75

correspond to the decision of an individual late consumer and the columns tothe decision of all the other late consumers. (Note: this is not a 2 × 2 game;the choice of column represents the actions of all but one late consumer.)The ordered pairs are the payoffs for the distinguished late consumer (the firstelement) and the typical late consumer (the second element).

Run No Run

Run (rx + y , rx + y) (c1, c2)No Run (0, rx + y) (c2, c2)

It is clear that if

0 < rx + y < c1 < c2

then (Run, Run) is an equilibrium and (No Run, No Run) is also anequilibrium.

The preceding analysis (of a bank run) is predicated on the assumption thatthe bank liquidates all of its assets in order to meet the demand for liquidity atdate 1. Thismay be the result of legal restrictions. For example, bankruptcy lawor regulations imposed by the banking authority may require that if any claimis not met, the bank must wind up its business and distribute the liquidatedvalue of its assets to its creditors. Some critics of the Diamond–Dybvig modelhave argued that bank runs can be prevented by suspension of convertibility. Ifbanks commit to suspend convertibility (i.e. they refuse to allow depositors towithdraw), once the proportion of withdrawals is equal to the proportion ofearly consumers, then late consumers will not have an incentive to withdraw. Alate consumer knows that the bank will not have to liquidate the long asset andwill have enough funds to pay him the higher promised amount in the secondperiod. If such an agent were to join the run on the bank in the middleperiod, he would be strictly worse off than if he waited to withdraw at thelast date.

To answer the criticism that suspension of convertibility solves the bank-runproblem,Diamond andDybvig (1983) proposed a sequential service constraint.Under this assumption, depositors reach the bank’s teller one after anotherand withdraw c1 until the bank is unable to meet any further demand. Thesequential service constraint has two effects. It forces the bank to deplete itsresources and it gives depositors an incentive to run early in hopes of beingat the front of the queue. The bank cannot use suspension of convertibility toprevent runs since it does not find out a run is in progress until it is too lateto stop it.


Another point to note about the suspension of convertibility is that it solvesthe bank-run problem only if the bank knows the proportion of early con-sumers. If the proportion of early consumers is random and the realizationis not known by banks, the bank cannot in general implement the optimalallocation by using suspension of convertibility.

3.6 EQUILIBRIUM BANK RUNS

The analysis offered byDiamond andDybvig pinpoints the fragility of bankingarrangements based on liquid liabilities and illiquid assets, but it does notprovide a complete account of equilibrium in the banking sector. Instead, itassumes that the bank’s portfolio (x , y) and deposit contract (c1, c2) are chosenat date 0 in the expectation that the first-best allocation will be achieved. Inother words, the bank run at date 1, if it occurs, is entirely unexpected at date 0.Taking the decisions at date 0 as given, we can define an equilibrium at date 1in which a bank run occurs; but this is not the same thing as showing that thereexists an equilibrium beginning at date 0 in which a bank run is expected tooccur. If banks anticipated the possibility of a bank run, their decisions at date0 would be different and that in turn might affect the probability or even thepossibility of a bank run at date 1. What we need is an equilibrium account ofbank runs that describes consistent decisions at all three dates. In this section,we provide a coherent account of bank runs as part of an equilibrium thatincludes the decisions made at date 0. We proceed by establishing a numberof facts or properties of equilibrium bank runs before describing the overallpicture.

The impossibility of predicting bank runs

The first thing we need to notice in constructing an equilibrium account ofbank runs is that a bank run cannot occur with probability one. If a bank run iscertain at date 0, the bank knows that each unit of the good invested in the longasset will be worth r units at date 1. If r < 1, the long asset is dominated bythe short asset and the bank will not invest in the long asset at all. If r = 1, thetwo assets are for all intents and purposes identical. In either case, the optimaldeposit contract is (c1, c2) = (1, 1) and there is no motive for a bank run: a lateconsumer will get the same consumption whether he joins the run or not. So,the best we can hope for is a bank run that occurs with positive probability.

The role of sunspots

The uncertainty of the bank run introduces a new element in our theory. Inthe current model, there is no uncertainty about aggregate fundamentals, such

3.6 Equilibrium Bank Runs 77

as asset returns, the proportion of early consumers, and so on. The kind ofuncertainty we are contemplating here is endogenous, in the sense that it is notexplained by shocks to the fundamentals of the model. How can we explainsuch uncertainty? Traditional accounts of bank runs often referred to “mobpsychology.” Modern accounts explain it as the result of coordination amongindividuals that is facilitated by extraneous variables called “sunspots.” Weshall have more to say about the distinction between these different types ofuncertainty in Chapter 5. For the moment, it is enough to note that the uncer-tainty is not explained by exogenous shocks, but is completely consistent withthe requirements of equilibrium, namely, that every individual is maximizinghis expected utility and that markets clear.

We begin by hypothesizing that a bank run occurs at date 1 with probability0 < π < 1. To be more concrete, we can assume there is some randomvariable (sunspot) that takes two values, say, high and low, with probabilitiesπ and 1 − π , respectively. When the realization of the random variable ishigh, depositors run on the bank and when it is low, they do not. Note that therandom variable has no direct effect on preferences or asset returns. It ismerelya device for coordinating the decisions of the depositors. It is rational for thedepositors to change their behavior depending on the value of the sunspotmerely because they expect everyone else to do so.

The bank’s behavior when bank runs are uncertain

The expectation of a bank run at date 1 changes the bank’s behavior at date0. As usual, the bank must choose a portfolio (x , y) and propose a depositcontract (c1, c2), but it does so in the expectation that the consumption stream(c1, c2) will be achieved only if the bank is solvent. In the event of a bank run,on the other hand, the typical depositor will receive the value of the liquidatedportfolio rx + y at date 1. This means that

• with probability π there is a bank run and the depositor’s consumption isrx + y , regardless of his type;

• with probability (1−π)λ there is no run, the depositor is an early consumer,and his consumption at date 1 is c1;

• and with probability (1 − π)(1 − λ) there is no run, the depositor is a lateconsumer, and his consumption at date 2 is c2.

The outcomes of the bank’s decisions when runs are anticipated are illustratedin Figure 3.3.

The optimal portfolio

If c1 denotes the payment to early consumers when the bank is solvent and(x , y) denotes the portfolio, then the expected utility of the representative


1 – π

π

1 – λ

λ

U(1)

U(c1*)

U(c2*)

Figure 3.3. Equilibrium outcome when runs are anticipated with probability π > 0.

depositor can be written

πU (y + rx) + (1 − π) {λU (c1) + (1 − λ)U (c2)} .Now suppose that we increase y by a small amount ε > 0 and decrease xby the same amount. We increase λc1 by ε and reduce (1 − λ) c2 by Rε. Thisinsures the feasibility constraints are satisfied at each date. Then the change inexpected utility is

πU ′(y + rx) (1 − r) ε + (1 − π){U ′(c1) − U ′ (c2)R

}ε + o(ε).

The optimal portfolio must therefore satisfy the first-order condition

πU ′(y + rx) (1 − r) + (1 − π)U ′(c1) = (1 − π)U ′ (c2)R.

If π = 0 then this reduces to the familiar conditionU ′(c1) = U ′ (c2)R. Theserelations are graphed in Figure 3.4. The latter condition holds at y∗ whilethe former holds at y∗∗. Thus, the possibility of a run increases the marginalvalue of an increase in y (the short asset has a higher return than the long assetin the bankruptcy state if r < 1) and hence increases the amount of the shortasset held in the portfolio.

The optimal deposit contract

Our next task is to show that a bank run is possible when the deposit contractis chosen to solve the bank’s decision problem. To maximize expected utility,the bank must choose the deposit contract (c1∗, c2∗) to satisfy the first-ordercondition

U ′(c1∗) = RU ′(c2∗). (3.11)


U�

y* y**

πU�(y + rx)(1 – r ) + (1 – π)U�(c1)

(1 – π)U�(c2)R

(1 – π)U�(c1)

Figure 3.4. The determination of the optimal portfolio when bank runs are possible.

This condition, which is familiar from our characterization of the first best,plays a crucial role in determining whether the bank is susceptible to runs.

Aswe saw earlier, if relative risk aversion is greater thanone, then the solutionof the first-order condition (3.11) must satisfy the inequality

c1 > 1.

This condition implies there exists the possibility of a run. If all the depositorstry to withdraw at date 1, the total demand for consumption is c∗1 > 1 butthe maximum that can be provided by liquidating all of the long asset is 1.However, there will be nothing left at date 2 so the depositors are better offjoining the run than waiting until date 2 to withdraw.

In what follows, we assume that the agent’s preferences satisfy the condi-tion that

• relative risk aversion is greater than one, that is,

−U′′(c)cU ′(c)

> 1, ∀c > 0.

To simplify the characterization of the equilibrium, we only consider thespecial case in which the long asset, when liquidated prematurely, yields asmuch as the short asset. In other words,

• the liquidation value of the long asset is r = 1.


This implies that the long asset dominates the short asset so, without essentialloss of generality, we can assume in what follows that the entire bank portfoliois invested in the long asset.

In the event of a bank run, the liquidated value of the bank’s portfolio isone unit of the good, so every depositor’s consumption is also one unit of thegood. If the bank is solvent, the depositors receive the promised consumptionprofile (c1, c2). Since these quantities only apply in the event that the bank issolvent, they are chosen to maximize the typical consumer’s expected utility inthe event that the bank is solvent. The deposit contract must solve the decisionproblem

max λU (c1) + (1 − λ)U (c2)s.t. Rλc1 + (1 − λ) c2 ≤ R.

To see why the budget constraint takes this form, note that the bank haspromised a total of λc1 units to the early consumers and this requires the bankto liquidate λc1 units of the long asset at date 1. The amount of the long assetleft is (1 − λc1) and this produces R (1 − λc1) units of consumption at date 2.Thus, themaximumamount that can be promised to late consumers (1 − λ) c2must be less than or equal to R (1 − λc1). In effect, one unit of consumptionat date 1 is worth R units of consumption at date 2.

Equilibrium without runs

So far we have assumed that a run occurs with probability π and that the banktakes this possibility as given in choosing an optimal deposit contract; however,the bank can avoid a run by choosing a sufficiently “safe” contract. Rememberthat our argument for the existence of a run equilibrium at date 1 was basedon the assumption that c1 > 1. Thus, if all the late consumers join the run onthe bank at date 1 there is no way that the bank can provide everyone withc1. In fact, the bank will have to liquidate all its assets and even then can onlygive each withdrawer 1, the liquidated value of its portfolio. More importantly,since the bank’s assets are exhausted at date 1, anyone waiting until date 2 towithdraw will receive nothing.

In order to remove this incentive to join the run, the bank must choose adeposit contract that satisfies the additional constraint c1 ≤ 1. If we solve theproblem

max λU (c1) + (1 − λ)U (c2)s.t. Rλc1 + (1 − λ) c2 ≤ R

c1 ≤ 1


we find the solution(c∗∗1 , c∗∗

2

) = (1,R). In this case, the bank will be able togive everyone the promised payment c1 at date 1 and if any late consumerswait until date 2 to withdraw there will be enough left over to pay them at leastR > 1. More precisely, if 1 − ε of the depositors withdraw at date 1, the bankhas to liquidate 1 − ε units of the long asset, leaving ε units of the long assetto pay to the remaining late consumers. Then each consumer who withdrawsat date 2 will receive εR/ε = R > 1.

A characterization of regimes with and without runs

If the bank anticipates a run with probability π , then with probability π thedepositor’s consumption is 1, regardless of his type. With probability 1 − π

there is no run and with probability λ the depositor is an early consumer andhis consumption is c∗1 and with probability 1− λ he is a late consumer and hisconsumption is c∗2 . The possible outcomes are illustrated in Figure 3.3 (above).The expected utility of the typical depositor will be

πU (1) + (1 − π){λU (c∗1 ) + (1 − λ)U

(c∗2)}

,

and we have shown that the bank’s choice of portfolio(x , y) = (1, 0) and

deposit contract(c∗1 , c∗2

)will maximize this objective, taking the probability π

of a run as given.Alternatively, if the bank chooses a deposit contract that avoids all runs, the

expected utility of the typical depositor is

λU (c∗∗1 ) + (1 − λ)U (c∗∗

1 ) = λU (1) + (1 − λ)U (R).

Whether it is better for the bank to avoid runs or accept the risk of a run withprobability π depends on a comparison of the expected utilities in each case.Precisely, it will be better to avoid runs if

πU (1) + (1 − π){λU (c∗1 ) + (1 − λ)U

(c∗2)}

> λU (1) + (1 − λ)U (R).

Notice that the left hand side is a convex combination of the depos-itors’ utility U (1) when the bank defaults and their expected utilityλU (c∗1 ) + (1 − λ)U

(c∗2)when the bank is solvent. Now, the expected utility

from the safe strategy λU (1) + (1 − λ)U (R) lies between these two values:

U (1) < λU (1) + (1 − λ)U (R)

< λU (c∗1 ) + (1 − λ)U(c∗2).


π0 π

λU(1) + (1 – λ)U(R )

πU(1) + (1 – π){λU(c1*) + (1 – λ)U(c2*)

Figure 3.5. Determination of the regions of π supporting (respectively not support-ing) runs.

So there exists a unique value 0 < π0 < 1 such that

π0U (1) + (1 − π0){λU (c∗1 ) + (1 − λ)U

(c∗2)} = λU (1) + (1 − λ)U (R)

and the bank will be indifferent between the two strategies if π = π0. Obvi-ously, the bank will prefer runs if π < π0 and will prefer no runs if π > π0.These two regions are illustrated in Figure 3.5.

We have shown that as long as the probability of a bank run is sufficientlysmall, there will exist an equilibrium in which the bank is willing to risk a runbecause the cost of avoiding the run outweighs the benefit. In that case, therewill be a run if the sunspot takes the high value and not otherwise. There isan upper bound (less than one) to the probability of a run, however. If theprobability of a run is too high, the bank will take action to discourage a runand the depositors will find it optimal to withdraw at the correct date.

Note that we have not specified what the sunspot is. It could be any publiclyobserved random variable that takes on a particular value with probabilityπ < π0. If such a variable exists, then depositors can in principle coordinateon this variable to support an equilibrium bank run.

3.7 THE BUSINESS CYCLE VIEW OF BANK RUNS

The previous sections have outlined a Diamond–Dybvig style account of bankruns in which extrinsic uncertainty plays a crucial role. Runs occur in this

3.7 The Business Cycle View of Bank Runs 83

framework because of late consumers’ beliefs. If all the late consumers believethere will be a run, they will all withdraw their money in the middle period.If they do not believe a run will occur, they will wait until the last period towithdraw. In both cases, beliefs are self-fulfilling. In the last section we usedthe terminology of sunspots to explain how coordination occurs. Traditionalaccounts of bank runs often referred to “mob psychology” as the motive forthe bank run or “panic.” This view of bank runs as panics has a long historybut it is not the only view. An alternative view of bank runs is that they are anatural outgrowth of weak fundamentals arising in the course of the businesscycle. An economic downturn will reduce the value of bank assets, raisingthe possibility that banks will be unable to meet their commitments in thefuture. If depositors receive information about an impending downturn in thecycle, they will anticipate financial difficulties in the banking sector and tryto withdraw their funds. This attempt will precipitate the crisis. According tothis interpretation, crises are not random events, but a rational response tounfolding economic circumstances. In other words, they are an integral partof the business cycle.

In Chapter 1 we briefly discussed financial crises in the US during theNational Banking Era from 1865 to 1914. Gorton (1988) conducted an empir-ical study to differentiate between the sunspot view and the business cycle viewof banking crises using data from this period. He found evidence consistentwith the view that banking panics are predicted by the business cycle. It isdifficult to reconcile this finding with the notion of crises as “random” events.Table 3.1 shows the recessions and crises that occurred in the US during theNational Banking Era. It also shows the corresponding percentage changes inthe currency/deposit ratio and the change in aggregate GDP, as proxied bythe change in pig iron production during these periods. The five worst reces-sions, as measured by the change in pig iron production, were accompanied bycrises. In all, crises occurred in seven of the eleven cycles. Using the liabilitiesof failed businesses as a leading economic indicator, Gorton finds that criseswere predictable events: whenever this leading economic indicator reached acertain threshold, a panic ensued. The stylized facts uncovered by Gorton thussuggest that, at least during the US National Banking Era, banking crises wereintimately related to the business cycle rather than some extraneous randomvariable. Calomiris and Gorton (1991) consider a broad range of evidencefrom this period and conclude that the data do not support the “sunspot” viewthat banking panics are random events. Among other things, they find that forthe five episodes they focus on, stock prices fell by the largest amount by farduring the pre-panic periods.

In this section, we adapt our model to allow us to consider the fundamentalor business cycle view of banking crises. In particular, instead of assuming the


Table 3.1. National Banking Era panics.

NBER CyclePeak–Trough

Panic date %�(Currency/deposit)∗

%� Pigiron†

Oct. 1873–Mar. 1879 Sep. 1873 14.53 −51.0Mar. 1882–May 1885 Jun. 1884 8.80 −14.0Mar. 1887–Apr. 1888 No Panic 3.00 −9.0Jul. 1890–May 1891 Nov. 1890 9.00 −34.0Jan. 1893–Jun. 1894 May 1893 16.00 −29.0Dec. 1895–Jun. 1897 Oct. 1896 14.30 −4.0Jun. 1899–Dec. 1900 No Panic 2.78 −6.7Sep. 1902–Aug. 1904 No Panic −4.13 −8.7May 1907–Jun. 1908 Oct. 1907 11.45 −46.5Jan. 1910–Jan. 1912 No Panic −2.64 −21.7Jan. 1913–Dec. 1914 Aug. 1914 10.39 −47.1

∗ Percentage change of ratio at panic date to previous year’s average.† Measured from peak to trough.(Adapted from Table 1, Gorton 1988, p. 233).

long asset has a certain return, we assume that the return is risky. Here we arefollowing the approach developed in Allen and Gale (1998) (cf. also Bryant1980).

• The long asset is a constant returns to scale technology that takes one unitof the good at date 0 and transforms it into RH units of the good at date2 with probability πH and RL units with probability πL . If the long asset isprematurely liquidated, one unit of the asset yields r units of the good atdate 1. We assume that

RH > RL > r > 0.

An intermediary takes a deposit of one unit from the typical consumer andinvests it in a portfolio consisting of y units of the safe, short asset and x unitsof the risky, long asset, subject to the budget constraint

x + y ≤ 1.

In exchange, the intermediary offers the consumer a contract promising c1units of consumption if he withdraws at date 1 and c2 units of consumptionif he withdraws at date 2. As before we assume that the intermediary cannotobserve the consumer’s type (i.e. early or late) and so cannotmake the contractcontingent on that. Amore stringent requirement is that the intermediary can-notmake the deposit contract contingent on the state of nature or, equivalently,the return to the risky asset.


Free entry and competition among the intermediaries leads them tomaximize the expected utility of their customers. This implies that the inter-mediaries will receive zero profits in equilibrium. In particular, this requiresthat the consumers receive the entire value of the remaining assets at date 2.Because the terminal value of the assets is uncertain, the intermediary willpromise a large amount that will certainly exhaust the value of the assets atdate 2. Without loss of generality we put c2 = ∞ and, in what follows, wecan characterize the deposit contract by the single parameter c1 = d , where dstands for the face value of the deposit at date 1.

Introducing risk in the form of random asset returns does not eliminate theDiamond–Dybvig phenomenon of bank runs based on self-fulfilling expect-ations or coordination on sunspots. In fact, theDiamond–Dybvigmodel is justa special case of the current model with RH = RL . In order to distinguish thisaccount of bank runs, we simply rule out the Diamond–Dybvig phenomenonby assumption and consider only essential bank runs, that is, runs that cannotbe avoided. Loosely speaking, we assume that if there exists an equilibrium inwhich there is no bank run as well as one or more that have bank runs, thenthe equilibrium we observe is the one without a bank run rather than the onewith a bank run.

Suppose that the bank has chosen a portfolio (x , y) and a deposit contractd at date 0. At date 1 the budget constraint requires

λd ≤ yand we can assume, without loss of generality, that the intermediary alwayschooses (x , y) and d to satisfy this constraint. Otherwise, the intermediarywill always have to default and the value of d becomes irrelevant. Conse-quently, the consumption of the late consumers, conditional on no run, will begiven by

(1 − λ)c2s = Rs(1 − y) + y − λd .

This is consistent with no run if and only if c2s ≥ d or

d ≤ Rs(1 − y) + y .This last inequality is called the incentive constraint. If this inequality is satisfied,there is an equilibrium in which late consumers wait until date 2 to withdraw.Since we only admit essential runs, the necessary and sufficient condition fora bank run is that the incentive constraint is violated, that is,

d > Rs(1 − y) + y .


Since RH > RL , this condition tells us that there can never be an essential runin state H unless there is also one in state L. There is no point choosing d solarge that a run always occurs, so we can restrict attention to cases in which arun occurs in state L if it occurs at all. There are then three different cases thatneed to be considered. In the first, the incentive constraint is never binding andbankruptcy is not a possibility. In the second case, bankruptcy is a possibilitybut the bank finds it optimal to choose a deposit contract and portfolio sothat the incentive constraint is (just) satisfied and there is no bankruptcy inequilibrium. In the third case, the costs of distorting the choice of depositcontract and portfolio are so great that the bank finds it optimal to allowbankruptcy in the low asset-return state.

Case I: The incentive constraint is not binding in equilibrium

In this case,we solve the intermediary’s decision problemwithout the incentiveconstraint and then check whether the constraint is binding or not. The inter-mediary chooses two variables, the portfolio y and the deposit contract d tomaximize the depositor’s expected utility, assuming that there is no bank run.With probability λ the depositor is an early consumer and receives d regardlessof the state. With probability 1 − λ, the depositor is a late consumer and thenhis consumption depends on the return to the risky asset. The total consump-tion in state s is equal to the return to the risky asset plus the remainder of thereturns from the safe asset after the early consumers have received their share,that is, Rs(1 − y) + y − λd . The consumption of a typical late consumer isjust this amount divided by the number of late consumers 1 − λ. Thus, theexpected utility is

λU (d) + (1 − λ)

{πHU

(RH (1 − y) + y − λd

1 − λ

)

+πLU

(RL(1 − y) + y − λd

1 − λ

)}.

This expression is maximized subject to the feasibility constraints 0 ≤ y ≤ 1and λd ≤ y .

Assuming that the optimal portfolio requires investment in both assets,i.e. 0 < y < 1, the optimal choice of (y , d) is characterized by the necessaryand sufficient first-order conditions. Differentiating the objective functionwith respect to d and taking account of the constraint λd ≤ y , the first-ordercondition for the deposit contract is

U ′(d) −{πHU

′(RH (1−y) + y −λd

1−λ

)+ πLU

′(RL(1−y) + y −λd

1−λ

)}≥ 0,


with equality if λd < y . Differentiating with respect to y and taking accountof the constraint λd ≤ y , the first-order condition for the portfolio is

πHU′(RH (1 − y) + y − λd

1 − λ

)(1 − RH )

+ πLU′(RL(1 − y) + y − λd

1 − λ

)(1 − RL) ≤ 0,

with equality if λd < y . If (y∗, d∗) denotes the solution to these inequalities,then (y∗, d∗) represents an equilibrium if the incentive constraint is satisfiedin state L:

d∗ ≤ RL(1 − y∗) + y∗.Let U ∗ denote the maximized value of expected utility corresponding to(y∗, d∗).

The consumption profile offered by the bank is illustrated in Figure 3.6,which shows the consumption at each date and in each state as a function of thereturn on the long asset. If the low state return RL is sufficiently high, say RL =R∗L , then the incentive constraint is never binding. The early consumers receivec1s = d = y/λ and the late consumers receive c2s = Rs

(1 − y) / (1 − λ) in

each state s = H , L.If the solution to the relaxed problem above does not satisfy the incen-

tive constraint, there are two remaining possibilities: either the intermediarychooses a contract that satisfies the incentive constraint, i.e. one that isconstrained by it, or the intermediary chooses a contract that violates the

ct (R)

c1(R)

c2(R)

0RL*** RL** RL* RH RS

Figure 3.6. Illustration of consumption in each period and state as a function of thelong asset’s return Rs for s = H , L.


incentive constraint in the low state, in which case there is default in thelow state.

Case II: The incentive constraint is binding in equilibrium

Suppose that (y∗, d∗) does not satisfy the incentive constraint. If the inter-mediary chooses not to default, the decision problem is to choose (y , d) tomaximize

λU (d) + (1 − λ) {πHU (cH ) + πLU (cL)}subject to the feasibility constraints 0 ≤ y ≤ 1 and λd ≤ y and subject to theincentive constraints

c2s = Rs(1 − y)+ y − λd

1 − λ≥ d , for s = H , L.

The incentive constraint will only bind in the low states s = L. Substitutingfor c2L = d , the expression for expected utility can be written as

λU (d) + (1 − λ)

{πHU

(RH (1 − y) + y − λd

1 − λ

)+ πLU (d)

},

where the assumption of a binding incentive constraint implies that

d ≡ RL(1 − y) + y .Since d is determined by the choice of y , the optimal contract is entirelydetermined by a single first-order condition. Substituting for d in the objectivefunction we obtain

{λ + (1 − λ)πL}U (RL(1 − y) + y)

+ (1 − λ)πHU

((RH − λRL)(1 − y) + (1 − λ)y

1 − λ

)

Note that the feasibility condition λd ≤ y must also be satisfied. We treatthis as a constraint while maximizing the objective function above. Then thefirst-order condition for y takes the form

{λ + (1 − λ)πL}U ′(d)(1 − RL)

+ (1−λ)πHU′(

(RH−λRL)(1−y) + (1−λ)y

1−λ

)(−RH+λRL+1−λ

1−λ

)≤ 0


or

(λ + (1 − λ)πL)U′(d)(1 − RL)

+ πHU′(RH (1 − y) + y − λd

1 − λ

)(−RH + λRL + 1 − λ) ≤ 0,

with equality if λd < y .Let (y∗∗, d∗∗) denote the solution to this problem and let U ∗∗ denote the

corresponding maximized expected utility.This case is also illustrated in Figure 3.6. If RL = R∗∗

L then the incentiveconstraint is binding and consumption is the same for early and late consumersin the low state: c1L = c2L = d = y + RL

(1 − y). In the high state, c1H = d

and c2H = RH(1 − y) / (1 − λ) as usual.

Case III: The incentive constraint is violated in equilibrium

Again, suppose that (y∗, d∗) does not satisfy the incentive constraint. If thereis default in the low state, the expected utility of the depositors is

πH

{λU (d) + (1 − λ)U

(RH (1 − y) + y − λd

1 − λ

)}+ πLU

(r(1 − y) + y) .

In this case, the first-order conditions that characterize the choice of d and ytake the form

πH

{λU ′(d) − λU ′

(RH (1 − y) + y − λd

1 − λ

)}≥ 0,

with equality if λd < y and

πHU′(RH (1− y) + y − λd

1− λ

)(1−RH ) + πLU

′ (r(1− y) + y) (1−RL) ≤ 0,

with equality if λd < y . Let(d∗∗∗, y∗∗∗) denote the solution to this problem

and U ∗∗∗ the corresponding maximized expected utility.This case is illustrated in Figure 3.6. If RL = R∗∗∗

L then bankruptcy occurs inthe low state and both early and late consumers receive the same consumption:c1L = c2L = y + RL

(1 − y) < d . In the high state, c1H = d and c2H =

RH(1 − y) / (1 − λ) as usual. This is an equilibrium solution only if

d∗∗∗ > RL(1 − y) + y ,


and

U ∗∗∗ > U ∗∗.

The first condition guarantees that the incentive constraint is violated,so that the intermediary must default in state L, and the second condi-tion guarantees that default is preferred to solvency.Otherwise the bank prefers(d∗∗, y∗∗) and there is no default.

3.8 THE GLOBAL GAMES APPROACH TO FINDING

A UNIQUE EQUILIBRIUM

Section 3.6 demonstrated how the sunspot approach allowed a completedescription of an equilibrium with bank runs. The weakness of this approachis that it does not explain why the sunspot should be used as a coordinationdevice. There is no real account of what triggers a crisis. This is particularly aproblem if there is a desire to use the theory for policy analysis.

Carlsson and van Damme (1993) showed how the introduction of a smallamount of asymmetric information could eliminate themultiplicity of equilib-ria in coordination games. They called the gameswith asymmetric informationabout fundamentals global games. Theirwork showed that the existence ofmul-tiple equilibria depends on the players having common knowledge about thefundamentals of the game. Introducing noise ensures that the fundamentalsare no long common knowledge and thus prevents the coordination that isessential to multiplicity. Morris and Shin (1998) applied this approach tomodels of currency crises. Rochet andVives (2004) andGoldstein and Pauzner(2005) have applied the same technique to banking crises. In this section wepresent a simple example of the global games approach provided by Allen andMorris (2001).

There are two depositors in a bank. Depositor i’s type is �i . If �i is less than 1,then depositor i is an early consumer and needs to withdraw his funds from thebank. If �i is greater than or equal to 1,he is a late consumer andhas no liquidityneeds. In this case he acts to maximize his expected return. If a depositorwithdraws his money from the bank, he obtains a guaranteed payoff of ω > 0.If both depositors keep their money in the bank then both obtain ρ where

ω < ρ < 2ω.

If a depositor keeps his money in the bank and the other depositor withdraws,he gets a payoff of 0.

3.8 The Global Games Approach to Finding a Unique Equilibrium 91

Note that there are four states of liquidity demand: both are early consumersandhave liquidity needs,depositor 1 only is an early consumer andhas liquidityneeds, depositor 2 only is an early consumer and has liquidity needs, and bothare late consumers and have no liquidity needs. If there is common know-ledge of fundamentals, and at least one depositor is an early consumer, theunique equilibrium has both depositors withdrawing. But if it is commonknowledge that both depositors are late consumers, they are playing a coord-ination game with the following payoffs. (The first element represents thepayoff to the player choosing the row strategy and the second element is thepayoff to the player choosing the column strategy.)

Remain Withdraw

Remain (ρ, ρ) (0,ω)

Withdraw (ω, 0) (ω,ω)

An important feature of this coordination game is that the total payoffs whenonly onepersonwithdraws early are less thanwhenbothpeoplewithdraw early.One set of circumstances where this would arise, for example, is when the bankcan close down after everybody has withdrawn, but when anybody keeps theirmoney in the bank then extra costs are incurred to stay open and the bank’sassets are dissipated more.

With common knowledge that neither investor is an early consumer, thisgamehas two equilibria: both remain and bothwithdraw.Wewill next considera scenario where neither depositor is an early consumer, both know that no oneis an early consumer, both know that both know this, and so on up to any largenumber of levels, but nonetheless it is not commonknowledge that no one is anearly consumer. We will show that in this scenario, the unique equilibrium hasboth depositors withdrawing. In other words, beliefs about others’ beliefs, orhigher-order beliefs as they are called, in addition to fundamentals, determinethe outcome.

Here is the scenario. The depositors’ types, �1 and �2, are highly correlated;in particular suppose that a random variable T is drawn from a smooth distri-bution on the non-negative numbers and each �i is distributed uniformly onthe interval [T − ε,T + ε], for some small ε > 0. Given this probability dis-tribution over types, types differ not only in fundamentals, but also in beliefsabout the other depositor’s fundamentals, and so on.

To see why, recall that a depositor is an early consumer if �i is less than 1.But when do both depositors know that both �i are greater than or equal to1 so they are late consumers? Only if both �i are greater than 1 + 2ε. This isbecause both players knows that the other’s �i is within 2ε of their own. For


example, suppose ε = 0.1 and depositor 1 has �1 = 1.1. She can deduce thatT is within the range 1.0−1.2 and hence that �2 is within the range 0.9−1.3.Only if �1 ≥ 1.2 does depositor 1 know that depositor 2 is a late consumer.

When do both investors know that both investors know that both �i aregreater than or equal to 1? Only if both �i are greater than 1 + 4ε. To see this,suppose that ε = 0.1 and depositor 1 receives �1 = 1.3. She can deduce thatT is within the range 1.2−1.4 and hence that depositor 2’s signal is within therange 1.1−1.5. However, if depositor 2 receives �2 = 1.1, then he sets a positiveprobability of depositor 1 having �1 within the range 0.9−1.3 as above. Onlyif depositor 1’s signal is greater or equal to 1 + 4ε would this possibility beavoided and both would know that both know that both are late consumers.

As we go up an order of beliefs the range goes on increasing. Hence it cannever be common knowledge that both depositors are late consumers and haveno liquidity needs.

What do these higher-order beliefs imply? It is simplest to consider whathappens when ε is very small. In this case, since T is smoothly distributed theprobability of the other depositor having an �i above or below approaches 0.5in each case as ε → 0 (see Figure 3.7).Wewill take it as 0.5 in what follows. (Analternative approach is to assume T is uniformly distributed in which case it isexactly 0.5 even away from the limit of ε → 0 – see Morris and Shin (2003).)

How do depositors behave in equilibrium? Observe first that each depositorwill withdraw if �i < 1 so the depositor is an early consumer. What about if�i ≥ 1?Given the structure of themodelwith a person being an early consumerwhen �i < 1 and a late consumer when �i ≥ 1, the most natural strategy fora depositor to follow is to choose a strategy of remaining only when �i > k

TT�–� T�+�T�

Pro

babi

lity

dens

ity fu

nctio

n of

T

Figure 3.7. The probability of the other depositor’s �i being above or below 0.5 asε −→ 0.

3.8 The Global Games Approach to Finding a Unique Equilibrium 93

for some k ≥ 1 and withdrawing otherwise. Suppose depositor 1 followsthis strategy. Consider what happens when �2 = k. Given our assumptionsabout ε being small and T being drawn from a smooth distribution, depositor2 deduces that there is a 0.5 probability that �1 < k and depositor 1 willwithdraw and a 0.5 probability that �1 ≥ 1 and that she will remain. Thepayoff of depositor 2 from remaining is

= 0.5 × ρ + 0.5 × 0 = 0.5ρ,

and the payoff from withdrawing is

= 0.5 × ω + 0.5 × ω = ω.

Since it is assumed that ρ < 2ω or equivalently that 0.5ρ < ω it follows thatdepositor 2 will also withdraw. In fact his unique best response is to withdrawif �2 is less than some cutoff point k∗ strictly larger than k where at k∗ theexpected payoffs from remaining and withdrawing are equated. Since the twodepositors are in symmetric positions, we can use the same argument to showthat depositor 1will have a cutoff point higher than k∗. There is a contradictionand both remaining cannot be an equilibrium. In fact the equilibrium forsmall ε is unique, with both agents always withdrawing. Given the other iswithdrawing, it is always optimal to withdraw.

The argument ruling out the equilibrium with both remaining depends onthe assumption ρ < 2ω. If this inequality is reversed then the same logic asabove can be used to show that the unique equilibrium has both remaining.Again the multiplicity is eliminated.

The arguments used above to eliminate an equilibrium rely on depositorsusing a switching strategy where below some level they withdraw and abovesome level they remain. Note that other types of equilibria have not been ruledout here. For a full analysis of global games, see Morris and Shin (2003).

Using a global games approach to ensure the uniqueness of equilibrium istheoretically appealing. It specifies precisely the parameter values for which acrisis occurs and allows a comparative static analysis of the factors that influ-ence this set. This is the essential analytical tool for policy analysis. However,what is really needed in addition to logical consistency is empirical evidencethat such an approach is valid. Currently there is a very limited empirical lit-erature. This is in the context of currency crises and is broadly consistent withthe global games approach (see Prati and Sbracia 2002; Tillman 2004; andBannier 2005).


3.9 LITERATURE REVIEW

There is a large literature on banking crises. Excellent surveys are provided byBhattacharya and Thakor (1993), Freixas and Rochet (1997), and Gorton andWinton (2003). This review will therefore be brief.

Bryant (1980) and Diamond and Dybvig (1983) developed the first modelsof banking crises. Both papers had consumers with random liquidity demandsand showed that deposit contracts allowed this risk to be insured. In Diamondand Dybvig (1983) bank runs were generated by sunspots while in Bryant(1980) they were the result of aggregate loan risk and asymmetric informa-tion about loan payoffs. Both papers were concerned with justifying depositinsurance. Diamond and Dybvig argue deposit insurance eliminates the badequilibrium without cost because it removes the incentives of late consumersto withdraw early so in equilibrium there are no costs to providing the insur-ance. In Bryant’s model, deposit insurance is desirable because it eliminatesincentives to gather costly information that is not socially useful.

Following Diamond and Dybvig, much of the literature on panic-basedruns was focused on the assumptions underlying their model. Cone (1983)and Jacklin (1987) pointed out that it was necessary for depositors to haverestricted trading opportunities, otherwise banks would have to competewith financial markets and this would eliminate the insurance they couldoffer.

As we have seen, the possibility of panic-based bank runs depends inan important way on the sequential service (or “first come-first served”)constraint. Without this, runs could be prevented by suspending convert-ibility. A number of papers sought to justify the existence of the sequentialservice constraint endogenously rather than appealing to legal restrictions.Wallace (1988) assumes that the fraction of the population requiring liquidityis random. He also assumes that agents are spatially separated from each otherbut are always in contact with the bank. These factors imply that a sequen-tial service constraint is optimal. Whereas Diamond and Dybvig were able toshow that deposit insurance was optimal, in Wallace’s model it is not. Build-ing on Wallace’s model, Chari (1989) shows that if the interbank market doesnot work well, because of regulatory restrictions of the type in place duringthe National Banking Era in the US, then banking panics can occur. With awell functioning interbank market, however, runs do not occur. Calomiris andKahn (1991) argue that the deposit contract, together with a sequential serviceconstraint, can be optimal because it provides an incentive for depositors tomonitor bank managers. Diamond and Rajan (2001) show that the possibilityof runs arising from demand deposits and a sequential service constraint can

3.9 Literature Review 95

be desirable if it ensures that banks will not renegotiate to extract more rentsfrom entrepreneurs that have borrowed from the bank.

The second category of crises involves aggregate risk arising from the busi-ness cycle. Bryant’s (1980) model falls in this category since he assumesaggregate loan risk and asymmetric information about the outcome of thisrisk to produce an incentive for some depositors to run. In Gorton’s (1985)model, depositors receive a noisy signal about the value of bank assets andif this suggests the value is low there is a panic. Banks that are solvent sus-pend convertibility and pay a verification cost to demonstrate this to investors.Chari and Jagannathan (1988) focus on a signal extraction problemwhere partof the population observes a signal about future returns. Others must thentry to deduce from observed withdrawals whether an unfavorable signal wasreceived by this group or whether liquidity needs happen to be high. Chari andJagannathan are able to show panics occur not only when the outlook is poorbut also when liquidity needs turn out to be high. Jacklin and Bhattacharya(1988) also consider a model where some depositors receive an interim sig-nal about risk. They show that the optimality of bank deposits compared toequities depends on the characteristics of the risky investment. Hellwig (1994)considers a model where the reinvestment rate is random and shows thatthe risk should be borne both by early and late withdrawers. Alonso (1996)demonstrates using numerical examples that contracts where runs occur maybe better than contracts which ensure runs do not occur because they improverisk sharing. As discussed above, Allen and Gale (1998) develop a model ofbusiness cycle risk with symmetric information where future prospects canbe observed by everybody but are not contractible. Runs occur when futureprospects are poor.

As Section 3.6 illustrated, one of the key issues in the bank-run literatureis that of equilibrium selection. Diamond and Dybvig appealed (among otherthings) to sunspots to act as the coordination device but did not modelthis fully. Postlewaite and Vives (1987) developed a model that does notrely on sunspots and that generates a unique equilibrium. In the contextof currency crises, Morris and Shin (1998) showed how the global gamesapproach of Carlsson and van Damme (1993) could be used to ensure equilib-rium is unique. Their approach links the panic-based and fundamental-basedapproaches by showing how the probability of a crisis depends on the fun-damentals. Morris and Shin (2003) provide an excellent overview of globalgames. Allen and Morris (2001) develop a simple example to show how theseideas can be applied to banking crises. Rochet andVives (2004) use the uniqueequilibrium resulting from their global games approach to undertake policyanalysis. They consider the role of ex ante regulation of solvency and liquidityratios and ex post provision of liquidity by the central bank. Goldstein and


Pauzner (2005) use the global games approach to show how the probability ofpanic-based runs can be made endogenous and related to the parameters ofthe banking contract.

There is a large empirical literature on banking crises, which will onlybe briefly touched on here. Sprague (1910) is the classic study of crises inthe National Banking Era. It was commissioned by the National MonetaryCommission after the severe crisis of 1907 as part of an investigation of thedesirability of establishing a central bank in the US. Friedman and Schwartz(1963) have written a comprehensive monetary history of the US from 1867to 1960. Among other things, they argue that banking panics can have severeeffects on the real economy. In the banking panics of the early 1930’s, bank-ing distress developed quickly and had a large effect on output. Friedmanand Schwartz argued that the crises were panic-based and offered as evidencethe absence of downturns in the relevant macroeconomic time series prior tothe crises. Gorton (1988) showed that banking crises in the National BankingEra were predicted by a leading indicator based on liabilities of failed busi-nesses. This evidence suggests banking crises are fundamental or businesscycle related rather than panic-based. Calomiris and Gorton (1991) provide awider range of evidence that crises are fundamental-based rather than panic-based. Wicker (1980, 1996) shows that, despite the absence of collapses in USnationalmacroeconomic time series, in the first two of the four crises identifiedby Friedman and Schwartz in the early 1930’s there were large regional shocksand attributes the crises to these shocks. Calomiris and Mason (2003) under-take a detailed econometric study of the four crises using a broad range ofdata and conclude that the first three crises were fundamental-based while thefourth was panic-based.


Banking crises have been an important phenomenon in many countries inmany historical periods. In this chapter we have developed a framework basedon Bryant (1980) and Diamond and Dybvig (1983) for analyzing these crises.There are two approaches that can be captured by the framework. The firstis crises that are based on panics. The second is crises that are based on poorfundamentals arising from the business cycle. There has been a significantdebate in the literature on which of these is the “correct” approach to take tocrises. As we have seen, there is evidence that both are empirically relevant.There is no need to confine attention to one or the other as is done in much ofthe literature. Both are important.

References 97

REFERENCES

Allen, F. and D. Gale (1998). “Optimal Financial Crises,” Journal of Finance 53,1245–1284.

Allen, F. and S.Morris (2001). “FinanceApplications of Game Theory,” in K. Chatterjeeand W. Samuelson (eds),Advances in Business Applications of Game Theory, Boston:Kluwer Academic Publishers, 17–48.

Alonso, I. (1996). “On Avoiding Bank Runs,” Journal of Monetary Economics 37, 73–87.Bannier, C. (2005). “The Role of Information Disclosure and Uncertainty in

the 1994/95 Mexican Peso Crisis: Empirical Evidence,” Review of InternationalEconomics, forthcoming.

Bhattacharya, S. and A. Thakor (1993). “Contemporary Banking Theory,” Journal ofFinancial Intermediation 3, 2–50.


Calomiris, C. and G. Gorton (1991). “The Origins of Banking Panics, Models, Facts,and Bank Regulation,” in R. Hubbard (ed), Financial Markets and Financial Crises,Chicago, IL: University of Chicago Press.

Calomiris, C. and C. Kahn (1991). “The Role of Demandable Debt in StructuringOptimal Banking Arrangements,”American Economic Review 81, 497–513.

Calomiris, C. and J. Mason (2003). “Fundamentals, Panics, and Bank Distress Duringthe Depression,”American Economic Review 93, 1615–1647.

Carlsson, H. and E. van Damme (1993). “Global Games and Equilibrium Selection,”Econometrica 61, 989–1018.

Chari, V. (1989). “Banking Without Deposit Insurance or Bank Panics: Lessons Froma Model of the U.S. National Banking System,” Federal Reserve Bank of MinneapolisQuarterly Review 13 (Summer), 3–19.

Chari, V. and R. Jagannathan (1988). “Banking Panics, Information, and RationalExpectations Equilibrium,” Journal of Finance 43, 749–60.

Cone, K. (1983). Regulation of Depository Institutions, PhD Dissertation, StanfordUniversity.


Diamond, D. and R. Rajan (2001). Liquidity Risk, Liquidity Creation and FinancialFragility: A Theory of Banking,” Journal of Political Economy 109, 2431–2465.

Freixas, X. and J. Rochet (1997).Microeconomics of Banking. Cambridge: MIT Press.Friedman, M. and A. Schwartz (1963). A Monetary History of the United States, 1867–1960, Princeton, NJ: Princeton University Press.

Goldstein, I. and A. Pauzner (2005). “Demand-Deposit Contracts and the Probabilityof Bank Runs,” Journal of Finance 60, 1293–1327.

Gorton,G. (1985). “Bank Suspension of Convertibility,” Journal of Monetary Economics15, 177–193.



Gorton, G. and A. Winton (2003). “Financial Intermediation,” in G. Constantinides,M. Harris, and R. Stulz (eds), Handbook of the Economics of Finance, Volume 1A,Chapter 8, Amsterdam: North Holland, 431–552.

Hellwig, M. (1994). “Liquidity Provision, Banking, and the Allocation of Interest RateRisk,” European Economic Review 38, 1363–1389.

Jacklin, C. (1987). “Demand Deposits, Trading Restrictions, and Risk Sharing,” inE. Prescott and N. Wallace (eds), Contractual Arrangements for Intertemporal Trade,Minneapolis, MN: University of Minnesota Press, 26–47.

Jacklin, C. and S. Bhattacharya (1988). “Distinguishing Panics and Information-Based Bank Runs: Welfare and Policy Implications,” Journal of Political Economy 96,568–592.

Kindleberger, C. (1978). Manias, Panics, and Crashes: A History of Financial Crises,New York: Basic Books.


Morris, S. and H. Shin (1998). “Unique Equilibrium in a Model of Self-FulfillingCurrency Attacks,”American Economic Review 88, 587–597.

Morris, S. and H. Shin (2003). “Global Games: Theory and Applications,” in M. Dewa-tripont, L. Hansen and S. Turnovsky (eds),Advances in Economics and Econometrics:Theory and Applications, Eighth World Congress Volume I, 56–114.

Postlewaite, A. and X. Vives (1987). “Bank Runs as an Equilibrium Phenomenon,”Journal of Political Economy 95, 485–491.

Prati, A. and M. Sbracia (2002). “Currency Crises and Uncertainty About Fundamen-tals,” IMF Working Paper 02/3.

Rochet, J. and X. Vives (2004). “Coordination Failures and the Lender of Last Resort:Was Bagehot Right after All?” Journal of the European Economic Association 2,1116–1147.

Sprague, O. (1910). A History of Crises Under the National Banking System, NationalMonetary Commission,Washington DC, US Government Printing Office.

Tillman, P. (2004). “Disparate Information and the Probability of Currency Crises:Empirical Evidence,” Economics Letters 84, 61–68.

Wallace, N. (1988). “Another Attempt to Explain an Illiquid Banking System: TheDiamond and Dybvig Model with Sequential Service Taken Seriously,” FederalReserve Bank of Minneapolis Quarterly Review 12 (Fall), 3–16.

Wicker, E. (1980). “A Reconsideration of the Causes of the Banking Panic of 1930,”Journal of Economic History 40, 571–583.

Wicker, E. (1996). The Banking Panics of the Great Depression, Cambridge: CambridgeUniversity Press.

4

Asset markets

In the last chapter, we examined the role of intermediaries as providers ofliquidity and risk sharing. We did so under the assumption that intermedi-aries operated in isolation. There were no other financial institutions and nofinancial markets. In the present chapter, by contrast, we restrict our attentionto asset markets and assume that there are no financial institutions. In thechapters that follow,we use the building blocks developed in these two chaptersto study economies in which financial intermediaries and financial marketscoexist and interact with each other. Financial markets allow intermediariesto hedge risks and to obtain liquidity by selling assets, but this can be a mixedblessing. In some contexts, markets allow intermediaries to achieve superiorrisk sharing, but in others they lead to increased instability. To understand howmarkets can destabilize financial intermediaries, we first need to understandthe relationship between market liquidity and asset-price volatility. This isan interesting topic in its own right. Its implications for the stability of thefinancial system will become clear in the next chapter.

4.1 MARKET PARTICIPATION

One of the most striking things about stock markets is the degree of pricevolatility. On any given day it is common for the largest movement of anindividual stock to be around 25 percent. The total market often moves byone or two percent. In October 1987 the market fell by around a third ina single day. These large changes in prices can trigger financial instability.This is particularly true in modern financial systems where many institutionsundertake complex risk management programs. Understanding asset pricevolatility is thus an important component of understanding financial crises. Inthis chapter we focus on asset markets alone. In the next chapter we will lookat the interaction of markets and financial institutions and see how the effectsinvestigated here can lead to fragility.

100 Chapter 4. Asset Markets

Why are stock prices so volatile? The traditional explanation is that pricevolatility is due to the arrival of new information about payoff streams anddiscount rates. There is a large body of evidence that suggests that informationis an important determinant of asset-price volatility (see, e.g. Fama 1970 andMerton 1987) but whether it is the only determinant is hotly debated. TheCrash of 1987 is an interesting example because stock prices fell by a largeamount despite the apparent absence of new information that day. Leroy andPorter (1981) and Shiller (1981) have argued that stock prices are characterizedby excess volatility : they are more volatile than the changes in payoff streamsand discount rates would predict. A number of authors have suggested that thedegree of excess volatility found by these studies can be attributed to the useof inappropriate econometric techniques (see Merton 1987 and West 1998 forsurveys of the literature); however, subsequentwork that avoids these problemsstill finds that there is excess volatility (see Campbell and Shiller 1988a,b andLeroy and Parke 1992).

So-called liquidity trading is another possible explanation for asset-pricevolatility. For a variety of reasons, financial institutions, firms, and individualshave sudden needs for cash and sell securities to meet such needs. If liquidityneeds are uncorrelated, one would expect them to cancel out in a large mar-ket, thus reducing their impact. Similarly, in a large market one mightexpect that the other traders would absorb a substantial amount of liquidity.In Sections 4.1–4.2 we develop a simple model of asset-price volatility inwhich small amounts of liquidity trading can cause significant price volatilitybecause the supply of liquidity in the market is also small.

The model in Sections 4.1–4.2 is based on complete participation, that is,every potential trader has unrestricted access to the market and participatesactively in it. There is extensive empirical evidence, however, that the assump-tion of complete participation is not justified. In Section 4.3 we examine theimplications of limited market participation. The fact is that most investors donot diversify across different classes of assets. For example, King and Leape(1984) analyze data from a 1978 survey of 6,010 US households with aver-age wealth of almost $250,000. They categorize assets into 36 classes and findthat the median number owned is eight. Mankiw and Zeldes (1991) find thatonly a small proportion of consumers hold stocks. More surprisingly, perhapseven among those with large liquid wealth, only a fairly small proportion ownstocks; of those with other liquid assets in excess of $100,000, only 47.7 percenthold stocks. Guiso et al. (2002) document the direct and indirect holdings ofstocks in a number of countries. Table 4.1 summarizes the results. It can beseen that equity holdings are increasing, particularly indirect holdings, in allthe countries. However, holdings of equity are low, particularly in Germanyand Italy, where the numbers of publically traded companies are small.

4.1 Market Participation 101

Table 4.1. Proportion of households investing in risky assets.

Year United UnitedStates Kingdom Netherlands Germany Italy

1983 19.1 8.9 n.a. 9.7 n.a.1989 16.8 22.6 n.a. 10.3 4.51995 15.2 23.4 11.5 10.5 4.01998 19.2 21.6 15.4 n.a. 7.3

Direct and indirectstockholding1983 n.a. n.a. n.a. 11.2 n.a.1989 31.6 n.a. n.a. 12.4 10.51995 40.4 n.a. 29.4 15.6 14.01998 48.9 31.4 35.1 n.a. 18.7

Source: Guiso et al. (2002), Table I.4, p. 9.

Other studies have found that investors’ diversification within equity port-folios is also limited. Blume et al. (1974) develop a measure of portfoliodiversification which takes into account the proportion of stocks held in indi-viduals’ portfolios. Using this measure, they find that the average amount ofdiversification is equivalent to having an equally weighted portfolio with twostocks. Blume and Friend (1978) provide more detailed evidence of this lack ofdiversification. They find that a large proportion of investors have only one ortwo stocks in their portfolios, and very fewhavemore than 10. This observationcannot be explained by the argument that investors are mainly holding mutualfunds. The reason is that in King and Leape’s (1984) study, only one percent ofinvestors’ wealth was in mutual funds compared to 22.3 percent held directlyin equities. The Blume et al. (1974) and Blume and Friend (1978) studies arefrom an earlier period when it is likely that the ownership of mutual funds wasan even smaller proportion of wealth, given the growth in mutual funds thathas occurred.

One plausible explanation of limited market participation is the fixed setupcost of participating in a market. In order to be active in a market, an investormust initially devote resources to learning about thebasic features of themarketsuch as the distribution of asset returns and so forth and how to monitorchanges through time. Brennan (1975) has shown that with fixed setup costsof this kind, it is only worth investing in a limited number of assets. Kingand Leape (1984) find evidence that is consistent with this type of model.Haliassos and Bertaut (1995), Bertaut (1998), and Vissing-Jorgensen (2002)present theoretical and empirical evidence that a fixed participation cost isconsistent with patterns of stock ownership.


Of course, it is not just individuals that we are interested in. Institutionsare responsible for investing a significant fraction of total wealth. Portfoliomanagers also face fixed costs of entering markets. The knowledge required tooperate successfully in markets suggests that a manager who tries to operatein a large number of markets will not perform as well as one who spe-cializes. In addition, agency problems limit wide participation in markets.Investors are concerned that institutions will take undesirable risks with theirmoney and so impose limits on the amount they can invest in particular classesof assets. Limited capital means that the amount that institutions trade withinany market they participate in is also limited. For simplicity, we will treatall investors, whether individual and institutional, as if they were individualinvestors investing their own money.

Limited participation by itself does not explain excess volatility. By defin-ition, a market is liquid if it can absorb liquidity trades without large changesin price. For asset-price volatility, what is needed is market illiquidity. In ourmodel, the liquidity of the market does not depend on the number of investorswho participate, that is, on the thickness or thinness of the market. On thecontrary, we assume that the market is always “thick;” in other words, it alwayshas a large number of traders. Instead, liquidity depends on the amount ofcash held by the market participants: this is the amount of cash that is avail-able at short notice to buy stocks from liquidity traders, investors who haveexperienced a sudden need for liquidity. If there is a lot of “cash in the market,”liquidity trades are easily absorbed and have little effect on prices. If there isvery little cash in the market, on the other hand, relatively small shocks canhave a large effect on prices.

The impact of market liquidity on asset pricing is seen most strikingly in theequilibrium pricing kernel. In equilibrium, the price of the risky asset is equalto the lesser of two amounts. The first amount is the standard discountedvalue of future dividends. The efficient markets hypothesis claims that theprice of a security is equal to the expected present value of the stream of futuredividends. This is true in our model as long as there is no shortage of liquidityin the market. When there is a shortage of liquidity, however, the asset price isdetermined by the amount of cash in themarket.More precisely, the asset priceis equal to the ratio of available liquidity to the amount of the asset supplied. Inthis case assets, we have cash-in-the-market pricing. Assets are “underpriced”and returns are excessive relative to the standard efficient markets formula.

The amount of cash in the market will depend on the second importantfeature of our theory, which is the participants’ liquidity preference. The higherthe average liquidity preference of investors in the market, the greater is theaverage level of the short asset in portfolios and the greater the market’s abil-ity to absorb liquidity trading without large price changes. Building on this

4.2 The Model 103

relationship between liquidity preference and asset-price volatility, we can seehow market participation helps determine the degree of volatility in the mar-ket. The amount of cash in the market and the amount of liquidity tradingboth depend on who decides to participate. Thus the participation decision isone avenue by which large endogenous changes in volatility can be effected, aswe shall see.

The main result in Section 4.3 is the existence of multiple Pareto-rankedequilibria. Inone equilibrium,with limitedparticipation,asset prices are highlyvolatile. In another (Pareto-preferred) equilibrium, participation is completeand asset prices are much less volatile.

4.2 THE MODEL

We make the usual assumptions of three dates, indexed by t = 0, 1, 2, and asingle good at each date. There are the usual two assets, a short asset and a longasset.

• The short asset is represented by a storage technology: one unit of the goodinvested in the short asset at date t is transformed into one unit of the goodat date t + 1, where t = 0, 1.

• The long asset is represented by a productive investment technology with atwo-period lag: one unit of the good invested in the long asset at date 0 istransformed into R > 1 units of the good at date 2.

The returns to both assets are assumed to be certain. The model can easilybe extended to allow for uncertainty about asset returns (see Allen and Gale1994), but here we want to focus on other sources of uncertainty.

The asset market contains a large number, strictly, a continuum,1 of ex anteidentical consumers. Each consumer is endowed with one unit of the good atdate 0 and nothing at dates 1 and 2. There is no consumption at date 0 andconsumers invest their endowment in a portfolio of long and short assets toprovide for future consumption at dates 1 and 2.

Each consumer learns at date 1 whether he is an early consumer who onlyvalues consumption at date 1 or a late consumerwho only values consumptionat date 2. If a consumer expects c1 units of consumption at date 1 when he

1 We represent the set of agents by the unit interval [0, 1], where each point in the intervalis a different agent. We normalize the measure of the entire set of agents to be equal to oneand measure the fraction of agents in any subset by its Lebesgue measure. The assumption of alarge number of individually insignificant agents ensures perfect competition, that is, no one hasenough market power to affect the equilibrium terms of trade.


is an early consumer and c2 units of consumption at date 2 when he is a lateconsumer, his utility will be a random variable

u(c1, c2) ={U (c1) with probability λ,U (c2) with probability 1 − λ

where the utility function U (c) has the usual neoclassical properties withU ′(c) > 0 and U ′′(c) ≤ 0. The probability of being an early consumer isdenoted by λ > 0. The only aggregate uncertainty concerns the demand forliquidity. We assume that λ is a random variable. For simplicity, suppose λ

takes two values:

λ ={

λH with probability π

λL with probability 1 − π

where 0 < λL < λH < 1.At date 0, individuals know the model and the prior distribution of λ. At

date 1, they observe the realized value of λ and discover whether they are earlyor late consumers.

4.3 EQUILIBRIUM

When a typical consumer makes his plans at date 0, he does not know whetherstateH or state L will occur and he does not know whether he will be an earlyor a late consumer; but he does know the probability of each of these eventsand rationally takes them into account in making his plans. More precisely, heknows that there are essentially four outcomes: he is either an early consumerin state H , a late consumer in state H , an early consumer in state L, or a lateconsumer in state L. Each state s = H , L occurs with probability 1/2 andthen, conditional on the state, the consumer becomes an early consumer withprobability λs and a late consumer with probability 1 − λs . The probabilitiesof the four outcomes are given in Table 4.2.

Table 4.2. Probability distribution ofindividual outcomes.

Early Late

State H 12λH

12 (1 − λH )

State L 12λL

12 (1 − λL)

4.3 Equilibrium 105

Although there is only a single good, we distinguish goods by the date andstate in which they are delivered. Since all consumption occurs at dates 1 and2 and there are two states of nature,H and L, there are four contingent com-modities, one for each ordered pair (t , s) consisting of a date t = 1, 2 and a states = H , L. A consumer will have a consumption bundle that consists of differ-ent quantities of all four contingent commodities. Let c = (c1H , c2H , c1L , c2L)denote the consumption bundle obtained in equilibrium, where cts denotesconsumption at date t in state s, that is, consumption of the contingent com-modity (t , s). Figure 4.1 illustrates the outcomes for the individual consumerand the consumption associated with each outcome.

The expected utility associated with a consumption bundle c =(c1H , c2H , c1L , c2L) is given by

1

2

{λHU (c1H ) + (1 − λH )U (c2H )

}+ 1

2

{λLU (c1L) + (1 − λL)U (c2L)

}.

If an individual is an early consumer and the state is H then he consumesc1H and gets utility U (c1H ). The probability of this happening is 1

2λH . Thefirst term is the expected utility from being an early consumer in state H . Theother terms are derived similarly. All of the consumer’s decisions are assumedto maximize the value of his expected utility.

The consumption bundle obtained by a typical consumer depends on hisportfolio decision at date 0 and the asset prices observed at future dates. Sup-pose the consumer invests x ≥ 0 units in the long asset and y ≥ 0 units in theshort asset, where x + y = 1. The consumer’s portfolio will produce y units ofthe good at date 1 and (1− y)R units of the good at date 2. An early consumer

t = 0

H

c1H

c2H

c2L

c1L

λH

L

λL

1 – λL

1 – λH

t = 1 t = 2

1/2

1/2

Figure 4.1. Information structure of the asset market model.


will want to consume as much as possible at date 1 and a late consumer willwant to consume as much as possible at date 2. We assume there is an assetmarket at date 1. The price of the long asset, measured in terms of that goodat date 1, is denoted by Ps in state s = H , L. The present value of the investor’sportfolio at date 1 in state s will be

w1s = y + Psx . (4.1)

Note that if the price of the long asset is Ps , then the implicit price of futureconsumption in terms of present consumption is ps ≡ Ps/R. The investor’swealth in terms of consumption at date 2 is

w1s

ps=(y

Ps+ x)R. (4.2)

Figure 4.2 illustrates the determination of a typical profile of consumptionat each date and state. Note that consumption is determined by the portfoliodecision at date 0, that is, the choice of x and y , and the asset prices in eachstate. Since consumers take prices as given, the consumer has determined hisconsumption (contingent on the state) once he has chosen the portfolio (x , y).

The determination of equilibrium for the asset market model can be brokendown into two steps. First, we can determine the asset prices in each state atdate 1 taking as given the portfolio decisions made at date 0. Then, for any pairof prices (PH , PL), we can determine the optimal portfolio at date 0, that is,the portfolio (x , y) that maximizes expected utility. To be certain that we havefound an equilibrium,we have to check that these two steps are consistent, that

t = 0

PH

c1H = y + PHx

c2H = (y/PH + x)R

c2L = (y/PL + x)R

c1L = y + PLx

PL

t = 1 t = 2

x + y = 1

Figure 4.2. Asset returns in the asset market model.

4.3 Equilibrium 107

is, that the portfolios chosen at date 0 will lead to the expected prices at date1 and that the expectation of those prices will lead to the choice of the sameportfolios at date 0. We begin with the analysis of market-clearing at date 1.

4.3.1 Market-clearing at date 1

Suppose that all the consumers have chosen the same portfolio (x , y) at date0. The budget constraint ensures that x+ y = 1 so in what follows we let 1− ydenote the investment in the long asset. Also, because the true state is knownand the analysis applies equally to both states, we can suppress the reference tothe state and let P denote the price of the long asset and λ the fraction of earlyconsumers.

The price P is determined by demand and supply. The supply of the longasset comes from the early consumers. An early consumer wants to liquidateall his assets in order to consume as much as possible at date 1. In particu-lar, the early consumers will inelastically supply their holdings of the shortasset, whatever the price. So there is a vertical supply curve and the quantitysupplied is

S = λ(1 − y)

because there are λ early consumers and each of them has 1 − y units of thelong asset.

The demand for the long asset comes from the late consumers, but thedetermination of demand is a little more subtle than the determination ofsupply. Because the late consumers do not want to consume until date 2, theyhave a non-trivial decision about which assets to hold between date 1 and date2. Because the true state is known at date 1 there is no uncertainty and, inparticular, the returns of the two assets are known for certain. One unit of thegood invested in the short asset at date 1 will produce one unit at date 2. Oneunit of the good will purchase 1/P units of the long asset at date 0 and this willproduce R/P units of the good at date 2. Consumers will hold whichever assethas the highest returns. There are three cases to be considered.

1. If R/P < 1, the return on the long asset is less than the return on the shortasset, and no one will want to hold the long asset between dates 1 and 2.

2. If R/P = 1, the one-period holding returns on the short and long assetsare equalized at date 1. Then late consumers should be indifferent betweenholding the two assets at date 1.


3. If R/P > 1, the one-period holding return on the long asset is greater thanthe return on the short asset and no one will want to hold the short assetbetween date 1 and date 2.

In Case 1, the demand for the long asset is zero. In Case 2, the demand forthe long asset is perfectly elastic, at least, up to the maximum amount that thelate consumers could buy. In Case 3, the late consumers will want to hold onlythe long asset, and they will supply their holdings of the short asset inelasticallyin exchange for the long asset. Since there are 1 − λ late consumers and eachholds y units of the short asset, the total supply of the short asset is (1 − λ)y .If the price of the long asset is P then the net demand for the long asset will be

D∗(P) = (1 − λ)y

P.

Thus the demand curve will have the form

D(P) =

0 if P > R,[0,D∗(R)] if P = R,D∗(P) if P < R.

Figure 4.3 illustrates the demand and supply curves for the long asset. There aretwo cases that are relevant. If S ≤ D∗(R), then the intersection of demand andsupply occurs where the demand curve is perfectly elastic and the equilibriumprice is P = R. If S > D∗(R) then the intersection occurs at the downward-sloping section of the demand curve and the price satisfies S = D∗(P).Substituting for the values of S and D∗(P) we see that P is determined bythe equation

λ(1 − y) = (1 − λ)y

P

or

P = (1 − λ)y

λ(1 − y) .

Putting together these two cases, we can see that

P = min

{R,

(1 − λ)y

λ(1 − y)}

. (4.3)

4.3 Equilibrium 109

This pricing formula illustrates the impact of liquidity on asset pricing. Whenliquidity is plentiful then the price of the long asset is simply the discountedpayoff where the discount rate is the opportunity cost given by the return onthe short asset.When liquidity is scarce the price of the long asset is determinedby the cash in the market. The early consumers exchange all of their holdingsof the long asset in exchange for the consumption goods given by the lateconsumers holdings of the short asset.

P

R

Qλ(1 – y)0

D*(P ) = (1 – λ)y/P

Figure 4.3. Demand for and supply of the long asset at date 1.

4.3.2 Portfolio choice

Now consider the investment decision of the consumers at date 0. Taking asgiven the asset prices PH and PL in each state at date 1, the investors will choosethe portfolio (y , 1 − y) to maximize their expected utility,

E

[λsU (ws) + (1 − λs)U

(wsps

)], (4.4)

where ws = y + Ps(1 − y) and ps ≡ Ps/R.An equilibrium consists of a pair of asset prices (PH , PL) and a portfolio

choice y such that prices are given by the equation (4.3) and the portfoliomaximizes (4.4) at the given prices.


4.4 CASH-IN-THE-MARKET PRICING

Using the model described above, we can show how liquidity preference affectsthe prices of assets.

No aggregate uncertainty

Consider first the case where there is no aggregate uncertainty, so λH = λL .Since the two states are identical we can reasonably assume that the asset priceis the same in each state, say, PH = PL = P . As we saw in the previous chapter,when there is no uncertainty about the asset price P , the only possible equi-librium value is P = 1. At any other price, one of the two assets is dominatedand will not be held at date 0, which implies that the asset market cannot clearat date 1.

When P = 1, the two assets have the same return at date 0 and the investor’swealth at date 1 is independent of his portfolio choice at date 0. If he is an earlyconsumer, his consumption at date 1 is

c1 = 1 − y + Py = 1

and, if he is a late consumer, his consumption at date 2 is

c2 =(1 − y + y

P

)R = R.

This consumption allocation is feasible for the economy if the averageinvestment in the short asset satisfies y = λ. Then

λc1 = λ = yand

(1 − λ) c2 = (1 − y)Ras required.

Aggregate uncertainty

Now suppose that there is aggregate uncertainty about the total demand forliquidity, as represented by fluctuations inλs . This implies non-zero asset-pricevolatility in the sense that

PH < PL .

4.4 Cash-in-the-Market Pricing 111

10.750.50.250

2

1.5

1

0.5

0

y

P

Figure 4.4. Prices as functions of y for parameters R = 1.2, λL = 0.5, λH = 0.67.

From the formula for prices (4.3) and the fact that λH > λL , we can see thatPH < PL unless PH = PL = R. This fact is illustrated in Figure 4.4.PH = PL = R implies that no one will ever hold the short asset at date 0,

because it is dominated by the long asset (one unit invested in the long assetis worth R > 1 units of the good at date 1). But if none of the short asset isheld at date 0, the price of the asset at date 1 will be zero. This contradictionimplies that the market-clearing prices must satisfy PH < PL . In fact, the shortis undominated only if PH < 1.

Comparative statics

Suppose that we have cash-in-the-market pricing in both states, that is,

Ps = (1 − λs)y

λs(1 − y)for states s = H , L. Then

PHPL

= (1 − λH )λL

λH (1 − λL). (4.5)

From this formula, we can derive a number of comparative static propertiesabout the effect of liquidity shocks on asset-price volatility.


To illustrate the possibilities it is helpful to consider a single parameter familyof liquidity shocks:

λH − 0.5 = 0.5 − λL = ε.

Then

PHPL

= (0.5 − ε)2

(0.5 + ε)2.

The solution is illustrated in Figure 4.5.

0.50.3750.250.1250

1

0.75

0.5

0.25

0

Pric

e vo

latil

ity

Figure 4.5. Price volatility PH/PL as a function of liquidity shock parameter ε.

Interpreting this diagram, we have to remember that it is based on theassumption that cash-in-the-market pricing holds in both states. As the dif-ference λH − λL increases, however, the ratio PH/PL converges to zero andit is more likely that PL = R will hold. Nonetheless, this gives a nice exam-ple of how the liquidity shocks can lead to very large fluctuations in assetprices.

To check these conclusions, we have calculated the complete equilibriumfor the special case in which consumers have logarithmic utility functionsU (c) = ln c . Table 4.3 shows the equilibrium values of prices and the portfoliowhen R = 1.2 and the two states are equiprobable.

When λH and λL are not too far apart (ε is small) we have cash-in-the-market pricing in both states. However, when the difference λH − λL becomes

4.4 Cash-in-the-Market Pricing 113

Table 4.3. The effect of aggregate uncertainty onprices; π = 0.5;R = 1.2;U (c) = ln(c).

λH λL PH PL y

0.5 0.5 1 1 0.50.51 0.49 0.961 1.041 0.50.53 0.47 0.887 1.128 0.50.545 0.455 0.834 1.200 0.50.6 0.4 0.830 1.2 0.5550.9 0.1 0.810 1.2 0.879

large enough (ε = 0.1), the price in the low state hits its upper bound PL = R.Investors increase the amount in the short asset and this increase in liquiditymeans that increases in price volatility are damped relative to the case wherethere is cash in the market pricing in both states.

The effect of changes in the demand for aggregate liquidity on price volatilityis determined by the the amount of cash in the market. If the amount of cash yheld by market participants is large relative to the variation in liquidity needsthen prices will not be very volatile. In the example described in Table 4.3, largeamounts of the short asset were held and this is why large changes in aggregateliquidity needswereneeded to cause significant price volatility. If the amount ofcash held by participants is small then even small changes in absolute liquidityneeds can have a significant effect on prices. The example in Table 4.4 illustratesthis phenomenon.

Table 4.4. The effect of aggregate uncertainty on prices withlimited cash in the market; π = 0.1; R = 1.2; U (c) = ln (c).

λH λL PH PL y

0.05 0.05 1 1 0.050.055 0.045 0.828 1.023 0.0460.06 0.04 0.687 1.052 0.0420.075 0.025 0.388 1.200 0.0300.09 0.01 0.384 1.2 0.037

In this example, we have assumed that the high state is much less likely thanthe low state. The effect of this change is to significantly lower the amount ofliquidity in the markets for similar values of ε. As a result, the effect of a givenabsolute change in liquidity demand on price volatility is much higher. Notice,however, that once we get to the point where PL = R then changes in aggregateuncertainty no longer affect price volatility very much.


Another interesting question is what happens when both λH and λL becomesmall. If the probability of being an early consumer is small in both states, weshould expect that consumers will tend to invest much more in the long assetand very little in the short asset. This raises the possibility that even very smallchanges in liquidity demand λH − λL can have a large effect on prices. Onceagain it is helpful to consider a one-parameter family of shocks

λs ={kε if s = Hε if s = L

for some k ≥ 1. Substituting these values in (4.5) we get

PHPL

= (1 − kε)εkε(1 − ε)

= (1 − kε)k(1 − ε)

.

Then

limε→0

PHPL

= 1

k.

Again, if k becomes very large, then the upper bound PL = R may be binding,but even so we see that the variation in prices can be very large even if thevariation in liquidity demand as measured by λH − λL = (k − 1) ε is verysmall.

4.5 LIMITED PARTICIPATION

We have shown in the previous section that what matters for price volatilityis not absolute changes in liquidity demand, but rather changes in liquiditydemand relative to the supply of liquidity. If a liquidity shock is large relative tothe supply of liquidity there is significant price volatility. This can be true evenif the liquidity shock is arbitrarily small. In this section we develop these ideasone step further. We introduce a fixed cost of participating in a market. Inorder to be active in a market, an investor must initially devote resourcesto learning about the basic features of the market such as the trading rules,the distribution of asset returns, and how to monitor changes through time.When there are costs of participating in the market, asset price volatility isdetermined by the supply of liquidity from the market participants rather thanfrom investors as a whole. If the market participants do not choose to supplymuch liquidity, the market will be characterized by high asset-price volatility,

4.5 Limited Participation 115

even if there is a willingness to supply liquidity elsewhere in the economy.This leads to the possibility of multiple equilibria, which differ in the extent ofmarket participation and market liquidity.

We assume there are two types of investor in the economy. Type-A investorsare aggressive, that is, they are more likely to participate in the market. Theyhave a low probability of being early consumers, hence a low preference forliquidity, and they also have low risk aversion compared to the second type.Type-B investors are bashful, that is, they are less likely to participate in themarket. They have a higher probability of being early consumers, hence ahigher preference for liquidity, and a higher degree of risk aversion.

When the cost of entering the market is sufficiently small, we can showthat there is always full participation. All investors enter the market, the aver-age amount of liquidity is high, and asset prices are not excessively volatile.As the cost of participation increases, new types of equilibria emerge. For highentry costs there is no participation. For intermediate entry costs there existsa limited-participation equilibrium, in which only the aggressive investors arewilling to enter the market. Because the market is dominated by investorswith low liquidity preference, holding small reserves of the short asset, evensmall variations in the proportion of liquidity traders can cause a signifi-cant variation in prices. There are highly liquid investors in the economy,holding large reserves of liquidity which could potentially dampen this volatil-ity, but they have chosen not to participate in this market, at least in theshort run.

The two types of equilibria react quite differently to small liquidity shocks.In the limited-participation equilibrium, with just one type of investor inthe market, there can be considerable price volatility even when shocks aresmall as shown in the previous section. In the full-participation equilibrium,the liquidity provided by the investors with high liquidity preference absorbssmall liquidity trades, so a small amount of aggregate uncertainty implies asmall amount of price volatility. Comparing the two equilibria, limited marketparticipation has the effect of amplifying price volatility relative to the full-participation equilibrium.

We can also generate multiple equilibria: for a non-negligible set of entrycosts, equilibria with full participation and with limited participation coexist.If asset prices are expected to be highly volatile, type-B investors will notparticipate. As a result, the market will be dominated by type-A investors whohold illiquid portfolios, which ensures that the market is illiquid and generatesthe expected volatility. In this way beliefs become self-confirming. On theother hand, if asset prices are expected to be stable, they will all participate,the market will be liquid because the average investor is holding a more liquidportfolio, and the expectation of stability becomes self-confirming.


The existence of multiple equilibria gives rise to the possibility of coordin-ation failure. Comparing equilibria, we see that one has lower volatility thanthe other. For some parameter values, these differences ensure that we canPareto-rank the two types of equilibria. Type-B investors are clearly betteroff when there is full participation, because they could have stayed out butchose to enter. Except when there are perverse income effects, the reduction involatility also benefits the type-A aggressive investors. In general, the fact thatthe prices of financial assets aremore volatile than other prices may ormay notbe socially undesirable. However, if there exists a Pareto-preferred equilibriumwith lower volatility, we can say that high volatility represents a market failure.

4.5.1 The model

We start with a brief description of the model. As before there are three datest = 0, 1, 2.

• There are two types of consumers/investors i = A,B and a continuum ofeach type. The total number of agents of type i is denoted Ni > 0.

• Consumers have the usual Diamond–Dybvig preferences, that is, they areearly consumerswith probabilityλi or late consumerswith probability 1−λi :

ui(c1, c2) ={Ui(c1) w. pr. λiUi(c2) w. pr. 1 − λi

where 0 < λi < 1 for i = A,B. They learn whether they are early or lateconsumers at date 1.

• Type A is assumed to be less risk averse than type B.• Each investor has an initial endowment of one unit of the good at date 0 and

nothing at future dates.• There is a single consumption good at each date and there are two assets:

– the short asset produces one unit of consumption at date t + 1 foreach unit invested at date t and the amount chosen by each investor isdenoted yi ;

– the long asset produces R > 1 units of consumption at date 2 for eachunit invested at date 0 and the amount chosen by investors of type i isdenoted xi .

• Investors can hold the short asset without cost. However, the long asset canonly be traded if an investor pays a fixed cost of e ≥ 0 utils and enters themarket for the long asset.


4.5.2 Equilibrium

Individual decisions

The decision tree faced by a typical investor is illustrated in Figure 4.6. Sincethe decisions are essentially the same for both types,we suppress the i subscriptin this section.

U{(y/PH + x)R} – e

U{(y/PL + x)R} – e

t = 0

PH

PL

U(y + PHx) – e

U(y + PLx) – e

t = 1 t = 2

In

EarlyLate

Early Late

U(1)

Out

High

Low

Figure 4.6. Consumer’s decision tree with participation decision.

At date 0, an investor first decides whether or not to enter the asset market. Ifhe decides not to enter, he can only invest in the short asset. Since the return onthe short asset is one, his consumption will be equal to his initial endowment,i.e. c = 1, and his utility will be U (1), whether he is an early or late consumer.If he decides to enter the asset market, he must pay the entry fee e (in utils)and then divides his endowment between an investment of x units in the longasset and y units in the short asset.

At date 1, the investor learns the true state of nature and whether he is anearly or late consumer. If he is an early consumer, he liquidates his portfolioand consumes the liquidated value. If the price of the long asset is Ps in states = H , L then the early consumer’s net utilitywill beU (y+Psx)−e. Otherwise,he is a late consumer and he rolls over his assets until the final period when he

consumes the liquidated value and receives a net utility U((

yPs

+ x)R)

− e.Given that the investor decides to enter the market, he chooses his portfolio(x , y)to maximize his expected utility, taking as given the prices PH and PL

that will prevail in each state. Using x = 1 − y , we can write his decision


problem as

U ∗(PH , PL) = max0≤y≤1

E

{λsU (y + Ps(1 − y)) − e

+(1 − λs)U

((y

Ps+ (1 − y)

)R

)}.

It is optimal for the investor to enter the market if

U ∗(PH , PL) ≥ U (1).

Market clearing at date 1

Let ni denote the number of investors of type i who enter the market and letyi denote the portfolio chosen by investors of type i who choose to enter. Themarket-clearing conditions are essentially the same as for the model with onetype. Since the argument is the same for both states, we suppress the subscripts in what follows.

The early consumers of both types will supply their holdings of the longasset inelastically. The total number of early consumers of type i is ni , thenumber of entrants, times λi , the fraction of early consumers. Since each earlyconsumer of type i supplies

(1 − yi

), the total supply of the long asset will be

S = nAλA(1 − yA

)+ nBλB (1 − yB).

If P = R then the late consumers are indifferent between holding the shortand the long asset between date 1 and date 2. If P < R then the long assetdominates the short asset and the late consumers will supply their holdings ofthe short asset inelastically in exchange for the long asset. The total demandfor the long asset will be

D∗(P) = nA (1 − λA) yA + nB (1 − λB) yBP

since there are ni(1 − λi) late consumers of type i and each can purchase yi/Punits of the long asset.

The price that satisfies S = D∗(P) is given by the ratio Y /S, where

Y = nA (1 − λA) yA + nB (1 − λB) yB

is the supply of the short asset by early consumers and S is the supply of thelong asset by late consumers. By the usual argument, the price of the long asset


is the minimum of R and Y /S, that is,

P = min

{R,Y

S

}.

This formula holds in each state. The difference in prices between the twostates is due to the different demands for liquidity, that is, the different valuesof λis for each type in each state.

Equilibrium conditions

To describe an equilibrium we need to specify the following elements:

• the entry decisions of the investors, represented by the numbers nA and nB ;• the portfolio choices of the investors, represented by the numbers yA and yB ;• and the equilibrium prices PH and PL .

As shown above, the equilibrium prices must satisfy the equation

Ps = min

{R,YsXs

}

where

Xs = nAλAs(1 − yA

)+ nBλBs (1 − yB)

and

Ys = nA (1 − λAs) yA + nB (1 − λBs) yB

for each s = H , L; the portfolio choice yi must solve

U ∗(PH , PL) = max0≤y≤1

E

{λisU (y + Ps(1 − y)) − e

+(1 − λis)U

((y

Ps+ (1 − y)

)R

)},

for each type i; and the entry decisions must be optimal in the sense that

U ∗i (PH , PL) > Ui(1) =⇒ ni = Ni

and

U ∗i (PH , PL) < Ui(1) =⇒ ni = 0.


4.5.3 Equilibrium with full participation

An equilibrium with full participation is one in which every investor choosesto enter the asset market, that is, nA = NA and nB = NB .

Suppose that entry into the asset market is free, that is, e = 0. Then fullparticipation is optimal. In fact, it is strictly optimal. To see this, consider thefollowing strategy. At date 0 the investor invests everything in the short assety = 1. At date 1 he consumes the returns to the short asset if he is an earlyconsumer; otherwise, he invests in the long asset at date 1 and consumes thereturns at date 2. The expected utility from this strategy is

1

2

{λiHU1(1) + (1 − λiH )Ui

(R

PH

)}+ 1

2

{λiLUi(1) + (1 − λiL)Ui

(R

PL

)}.

Since we know that Ps ≤ R for s = H , L the expected utility must be atleast U (1). However, we cannot have Ps = R for both states s = H , L, forthen the short asset would be dominated at date 0, no one would hold theshort asset, and the price of the long asset would be zero at date 1. Thus, inany equilibrium Ps < R for at least one state s = H , L. But this means thatconsumption is strictly greater than one for the late consumers in some state.Thus,U ∗

i (PH , PL) > Ui(1).If it is strictly optimal to enter the market when the entry cost is zero then it

must also be strictly optimal to enter the market for all sufficiently small entrycosts e > 0. Thus, we have proved the following proposition.

Proposition 1 For all e > 0, sufficiently small, an equilibrium{(nA , yA

),(

nB , yB), (PH , PL)

}must entail full participation, that is,

nA = NA and nB = NB .

4.5.4 Full participation and asset-price volatility

To illustrate further the properties of the full-participation equilibrium, weuse an example in which the investors of type A have a random liquidity shockand investors of type B have a non-random liquidity shock:

λAs ={k > 0 if s = H

0 if s = Land

λBH = λBL = λB .


When there is full participation the pricing formula requires Ps =min {R,Ys/Ss}. Now as k → 0, so λAH → 0, and type-A investors becomemore and more certain of being late consumers yA → 0 so

YsSs

→ (1 − λB)yBλBxB

= Q,

say, where Q is a constant since λB is a constant. In the limit,Ps = min{R, Q}.

Recall that R > 1 for all s. Then Q > 1 implies Ps > 1 for all s. This impliesthat the short asset is dominated by the long asset at date 0 so no one will holdthe short asset; but this is not consistent with equilibrium. On the other hand,if Q < 1 then Ps < 1 for all s. This means that the short asset dominates thelong asset at date 0, which is also inconsistent with equilibrium. Therefore,

Ps = Q = 1 for s = H , L.

Thus, as liquidity shocks among the type As become small, the type Bs are ableto absorb the shocks, which have no effect on prices.

Along the same lines, it can be seen that as NB → ∞, holding NA constant,Ps tends to the same limit, and a similar result holds. As the number of typeBs goes up, the amount of liquidity in the market becomes large relative to thefluctuations in liquidity trading, and this dampens volatility.

4.5.5 Limited participation and asset-price volatility

Continuing with the special case considered in the preceding section, supposethat there exists an equilibrium inwhich only typeA investors enter themarket.That is, nA = NA and nB = 0. With only type A in the market at date 1, therewill be no demand for the short asset in state L and PL = R. The price in statePH must satisfy the first-order condition that makes investors willing to holdboth assets at the margin in date 0. In the limit, as k → 0, investors of typeA are late consumers with probability one and invest all of their wealth in thelong asset. So, in the limit, a late consumer receives utility UA(R) regardless ofthe state. The marginal utility of income is also constant across states, so theinvestor behaves as if he is risk neutral. Then in order to be willing to holdboth assets, the expected returns must be equal over two periods, that is,

R = 1

2

R

PH+ 1

2.

The left hand side is the return to one unit invested in the long asset. The righthand side is the expected return to one unit invested in the short asset at date 0.


To see this, suppose that the investor holds his wealth in the form of the shortasset at date 0 and invests in the long asset at date 1 (this is always weaklyoptimal and will be strictly optimal if Ps < R). With probability π he is instate H at date 1 and can buy 1/PH units of the long asset at date 1, whichwill yield R/PH at date 2. With probability 1

2 he is in state L at date 1 and canbuy 1/PL = 1/R units of the long asset at date 1, which will yield R/R = 1 atdate 2. Taking the expected value of these returns gives us the right hand sideof the equation. We can solve this equation for the asset price in the high state:

PH = R

2R − 1< 1.

In this equilibrium, unlike the full-participation equilibrium, there is sub-stantial asset price volatility even in the limit as k → 0. If the asset-pricevolatility is sufficiently high and the type-B investors are sufficiently risk averse,it may be optimal and hence an equilibrium phenomenon, for the type-Binvestors to stay out of the market. This situation is illustrated in Figure 4.7.

UA(R)

0 e→

45º

UB*

UB**

e0 e1 e2

Figure 4.7. Investors’ entry decisions for different entry costs.

On the horizontal axis we measure the entry cost e and on the vertical axisthe expected utility of entering the asset market. Without loss of generality, wenormalize utilities so that UA(1) = UB(1) = 0. We have already seen that, inthe limit as k → 0, the expected utility of type A isUA(R). LetU ∗∗

B denote theexpected utility of type B, before taking account of the entry cost, in the limitedparticipation equilibrium as k → 0. Under the assumption that the utilityfrom staying out of the market is zero, an investor will enter the market if thegross expected utility U ∗∗

B is greater than the entry cost e. In other words, hewill enter if his payoff is above the 45◦ line. As long asU ∗∗

B < UA(R) as drawn,there will be a range of entry costs in which the type-A investors find it worth


while to enter and the type-B investors do not. For any e < e2, it is worthwhilefor the type-A investors to enter and, for any e > e0, it is not worthwhile for thetype-B investors to enter if they anticipate the high degree of volatility associatedwith the limited participation equilibrium. Thus, for any entry cost e1 < e < e2there will be an equilibrium with limited participation.

4.5.6 Multiple Pareto-ranked equilibria

Figure 4.7 also illustrates the possibility of multiple equilibria. Let U ∗B denote

the expected utility of the type B investors in the full participation equilibriumwhere asset-price volatility is extremely low. Again, U ∗

B is the gross expectedutility before the entry cost is substracted. If U ∗

B < UA(R), as shown in thediagram, then for any e < e1 it is worthwhile for both types to enter if theyanticipate the low asset-price volatility associated with the full-participationequilibrium. So there is a full-participation equilibriumcorresponding to entrycosts e < e1.

If type-B’s (gross) expected utility in the limited participation equilibriumU ∗∗B is less than the (gross) expectedutility in the full-participation equilibriumU ∗B , as shown in the diagram, then e0 < e1 and there will be an intervale0 < e < e1 where both kinds of equilibria, those with full participation andthose with limited participation, exist.

In fact, we can show that in the limit these equilibria are Pareto-ranked. Thetype-B investors always prefer the full-participation equilibrium. They havethe option of just holding the short asset, as they do in the limited participa-tion equilibrium, but choose not to do so. The reason the full-participationequilibrium is better for them is that it allows them to obtain the benefits ofthe long asset’s high return. As for the type-A investors, in the limit as k → 0,they obtain UA(R) − e in both equilibria and so are indifferent.

A general comparison of the type-A investors outside the limit as k → 0is more complex for the simple reason that there are both risk and incomeeffects. The risk effect arises from the fact that prices are more variable in thelimited-participation equilibrium than in the full-participation equilibrium.This tends to make the type-A investors prefer the full-participation equilib-rium. However, in addition, there is an income effect which can more thanoffset the risk effect.

In the limited-participation equilibrium, the type-A investors who are earlyconsumers trade with the type-A investors who are late consumers. Their ini-tial expected utility is an average over these two states. There are no transfersbetween the type-A investors and the type-B investors. In the full-participationequilibrium, however, λAs < λB so there are proportionately fewer type-A


investors than type-B investors who are early consumers, and more whoare late consumers. On average, the type-A investors buy more of the long-term asset from the type-B investors than they sell to them. The price ofthe long-term asset determines whether there is a transfer from the type-Ainvestors to the type-B investors or vice versa. It is possible that the transferof income between the type-A investors and the type-B investors in the full-participation equilibrium outweighs the risk effect and the type-A investorsprefer the limited-participation equilibrium. Thus, the equilibria cannot bePareto-ranked in general.

4.6 SUMMARY

The results in this chapter have been demonstrated for quite special examples.However, they hold for a fairly wide range of cases. Allen and Gale (1994)documents how the results hold if the return on the long asset R is randomand there is a continuous range of values concerning the uncertainty associatedwith the proportion of early consumers.

The first major result is to show that price volatility is bounded away fromzero as aggregate uncertainty about liquidity demands tends to zero. Whenliquidity is scarce, asset prices are not determined by discounted cash flows,but rather by the amount of liquidity in the market. It is the relative variationin the amount of the long asset and the liquidity supplied to the market that isimportant for price volatility. The absolute amounts of both fall as aggregateuncertainty becomes negligible, but the relative variation remains the same.

There is considerable empirical evidence suggesting that investors partici-pate in a limited number of markets because of transaction costs. We studiedthe interaction between liquidity preference and limited market participationand the effect of limited market participation on asset-price volatility. For arange of entry costs, there exist multiple equilibria with very different welfareproperties. Asset-price volatility can be excessive in the sense of being Pareto-inferior as well as in the sense of being unexplained by the arrival of newinformation.

REFERENCES

Allen, F. and D. Gale (1994). “Limited Market Participation and Volatility of AssetPrices,”American Economic Review 84, 933–955.

References 125

Bertaut, C. (1998). “Stockholding Behavior of U.S. Households: Evidence from the1983–1989 Survey of Consumer Finances,” Review of Economics and Statistics 80,263–275.

Blume, M., J. Crockett, and I. Friend (1974). “Stock Ownership in the United States:Characteristics and Trends,” Survey of Current Business 54, 16–40.

Blume, M. and I. Friend (1978). The Changing Role of the Individual Investor: ATwentieth Century Fund Report, New York: Wiley.

Brennan, M. (1975). “The Optimal Number of Securities in a Risky Asset Portfoliowhen there are Fixed Costs of Transacting: Theory and Some Empirical Results,”Journal of Financial and Quantitative Analysis 10, 483–496.

Campbell, J. and R. Shiller (1988a). “Stock Prices, Earnings and Expected Dividends,”Journal of Finance 43, 661–676.

Campbell, J. and R. Shiller (1988b). “The Dividend-Price Ratio, Expectations of FutureDividends and Discount Factors,”Review of Financial Studies 1, 195–228.

Fama, E. (1970). “Efficient Capital Markets: A Review of Theory and Empirical Work,”Journal of Finance 25, 383–417.

Guiso, L., M. Haliassos, and T. Jappelli (2002). Household Portfolios. Cambridge, MA:MIT Press.

Haliassos,M. and C. Bertaut (1995). “Why Do So Few Hold Stocks?”Economic Journal105, 1110–1129.

King, M. and J. Leape (1984). “Wealth and Portfolio Composition: Theory and Evi-dence,” National Bureau of Economic Research (Cambridge, MA), Working PaperNo. 1468.

LeRoy, S. and W. Parke (1992). “Stock Price Volatility: Tests Based on the GeometricRandom Walk,”American Economic Review 82, 981–992.

LeRoy, S. and R. Porter (1981).“The Present Value Relation: Tests Based on ImpliedVariance Bounds,” Econometrica 49, 555–574.

Mankiw, N. G. and S. P. Zeldes (1991). “The Consumption of Stockholders and Non-Stockholders,” Journal of Financial Economics 29, 97–112.

Merton, R. (1987). “On the Current State of the Stock Market Rationality Hypothesis,”in R. Dornbusch, S. Fisher, and J. Bossons (eds),Macroeconomics and Finance: Essaysin Honor of Franco Modigliani. Cambridge, MA: MIT Press, 99–124.

Shiller, R. (1981). “Do Stock Prices Move too Much to be Justified by SubsequentChanges in Dividends?”American Economic Review 71, 421–436.

Vissing-Jorgensen,A. (2002). “Towards an Explanation of Household Portfolio ChoiceHeterogeneity: Nonfinancial Income and Participation Cost Structures,” workingpaper, University of Chicago.

West, K. (1988). “Dividend Innovations and Stock Price Volatility,” Econometrica 56,37–61.

5

Financial fragility

In the preceding two chapters, we separately considered the operation of assetmarkets and intermediaries, such as banks. In the present chapter, we combinethese elements and begin the study of their interaction. From this we willgain important insight into the phenomenon of financial fragility. We use thephrase “financial fragility” to describe situations in which small shocks havea significant impact on the financial system. One source of financial fragilityis the crucial role of liquidity in the determination of asset prices. There aremany historical illustrations of this phenomenon. For example, Kindleberger(1978, pp. 107–108) argues that the immediate cause of a financial crisis

may be trivial, a bankruptcy, a suicide, a flight, a revelation, a refusal of credit to someborrower, some change of view which leads a significant actor to unload. Prices fall.Expectations are reversed. Themovement picks up speed. To the extent that speculatorsare leveraged with borrowed money, the decline in prices leads to further calls on themfor margin or cash, and to further liquidation. As prices fall further, bank loans turnsour, and one or more mercantile houses, banks, discount houses, or brokerages fail.The credit system itself appears shaky and the race for liquidity is on.

A particularly interesting historical example is the financial crisis of 1763documented by Schnabel and Shin (2004). The banks in those days were differ-ent from modern commercial banks. They did not take in deposits and makeloans. Instead the “bankers” were merchants involved in the trade of goodssuch as wheat (this is where the term “merchant banker” comes from). Theirprimary financial role was to facilitate payments between parties using billsof exchange. A bill was like an IOU in which one party acknowledged thathe had received a delivery of wheat, say, and promised payment at a specifiedfuture date. Sometimes reputable bankers could make their creditworthinessavailable to others, by allowing people known to them to draw bills on themin exchange for a fee. These bills could then be used as a means of paymentor to raise capital by selling the bill in the capital market. The widespread useof these bills in financial centers, such as Amsterdam and Hamburg, led tointerlocking claims among bankers.

The de Neufville brothers’ banking house in Amsterdam was one of themost famous in Europe at the time. The end of the Seven Years War led to an

Chapter 5. Financial Fragility 127

economic downturn that triggered the bankruptcy of their banking house inAmsterdam. This forced them to sell their stocks of commodities. In the shortrun, the liquidity of the market was limited and the commodity sales resultedin lower prices. The fall in prices in turn put other intermediaries under strainand forced them to sell. A collapse in commodity prices and a financial crisisfollowed with many merchant bankers forced out of business.

Recent examples provide a stark illustration of how small events can causelarge problems. As discussed in Chapter 1, in August 1998 the Russian govern-ment announced a moratorium on about 281 billion roubles ($13.5 billion)of government debt. Despite the small scale of the default, it triggered a globalcrisis and caused extreme volatility in many financial markets. The hedgefund Long Term Capital Management came under extreme pressure. DespiteLTCM’s small size in relation to the global financial system, the Federal ReserveBank of New York was sufficiently worried about the potential for a crisis ifLTCM went bankrupt that it helped arrange for a group of private banks topurchase the hedge fund and liquidate its positions in an orderly way. TheFed’s concern was that, if LTCM went bankrupt, it would be forced to liquidateall its assets quickly. LTCM held many large positions in fairly illiquid markets.In such circumstances, prices might fall a long way if large amounts were soldquickly. This could put strain on other institutions, which would be forced tosell in turn, and this would further exacerbate the problem, as Kindlebergerdescribes in the passage above.

In this chapter we will show how the interaction of financial intermediariesand markets can lead to financial fragility. In particular we will show that:

• Small events, such as minor liquidity shocks, can have a large impact on thefinancial system because of the interaction of banks and markets.

• The role of liquidity is crucial. In order for financial intermediaries to havean incentive to provide liquidity to a market, asset prices must be volatile.

• Intermediaries that are initially similar may pursue radically different strat-egies, both with respect to the types of asset they invest in and their risk ofdefault.

• The interaction of banks and markets provides an explanation for systemicor economy-wide crises, as distinct from models, such as Bryant (1980) andDiamond and Dybvig (1983), that explain individual bank runs.

The central idea developed in the rest of this chapter is that, when mar-kets are incomplete, financial institutions are forced to sell assets in order toobtain liquidity. Because the supply of and demand for liquidity are likely tobe inelastic in the short run, a small degree of aggregate uncertainty can causelarge fluctuations in asset prices. Holding liquidity involves an opportunitycost and the suppliers of liquidity can only recoup this cost by buying assets at

128 Chapter 5. Financial Fragility

firesale prices in some states of the world; so, the private provision of liquidityby arbitrageurs will always be inadequate to ensure complete asset-price sta-bility. As a result, small shocks can cause significant asset-price volatility. If theasset-price volatility is severe enough, banks may find it impossible to meettheir fixed commitments and a full-blown crisis will occur.

5.1 MARKETS, BANKS, AND CONSUMERS

The model used in this chapter combines intermediation and asset marketsand is based on that inAllen andGale (2004). Individual investors deposit theirendowments in banks in exchange for a standard deposit contract, promisingthem a fixed amount of consumption at dates 1 and 2. The banks use the assetmarket at date 1 to obtain additional liquidity or get rid of excess liquidity asneeded. As we shall see, there are also a number of small additional changescompared to the models of the preceding chapters.

There are three dates, t = 0, 1, 2; contracts are drawn up at date 0, and allconsumption occurs at dates 1 and 2. There is a single good at each date. Thegood can be used for consumption or investment.

• Investment in the short or liquid asset can take place at date 1 or date 2. Oneunit of the good invested in the short asset at date t yields one unit at datet + 1 for t = 0, 1.

• The long asset takes two periods to mature and is more productive thanthe short asset. Investment in the long asset can take place only at date 0.One unit invested at date 0 produces a certain return R > 1 units at date 2.Claims on the long asset can be traded at date 1.

There is the usual trade-off between liquidity and returns: long-term invest-ments have higher returns but take longer to mature (are less liquid). Bycontrast, there is no risk–return trade-off: we assume that asset returns arenon-stochastic in order to emphasize that, in the present model, financialcrises are not driven by shocks to asset returns.

There is a continuum of ex ante identical consumers, whose measure isnormalized to unity. Each consumer has an endowment (1, 0, 0) consistingof one unit of the good at date 0 and nothing at subsequent dates. Thereare two (ex post) types of consumers at date 1: early consumers, who valueconsumption only at date 1; and late consumers, who value consumption onlyat date 2. If λ denotes the probability of being an early consumer and ct denotesconsumption at date t = 1, 2, then the consumer’s ex ante utility is

E[u(c1, c2)] = λU (c1) + (1 − λ)U (c2).

5.1 Markets, Banks, and Consumers 129

U (·) is a neoclassical utility function (increasing, strictly concave, twicecontinuously differentiable).

It is important in what follows to distinguish between two kinds ofuncertainty. Intrinsic uncertainty is caused by stochastic fluctuations in theprimitives or fundamentals of the economy. An example would be exogenousshocks that affect liquidity preferences. Extrinsic uncertainty by definitionhas no effect on the fundamentals of the economy. An equilibrium with noextrinsic uncertainty is called a fundamental equilibrium, because the endoge-nous variables are functions of the exogenous primitives or fundamentalsof the model (endowments, preferences, technologies). An equilibrium withextrinsic uncertainty is called a sunspot equilibrium, because endogenous vari-ables may be influenced by extraneous variables (sunspots) that have no directimpact on fundamentals. A crisis cannot occur in a fundamental equilibriumin the absence of exogenous shocks to fundamentals, such as asset returns orliquidity demands. In a sunspot equilibrium, by contrast, asset prices fluctu-ate in the absence of aggregate exogenous shocks, and crises appear to occurspontaneously.

There are three sources of intrinsic uncertainty in the model. First, eachindividual consumer faces idiosyncratic uncertainty about her preferencetype (early or late consumer). Second, each bank faces idiosyncratic uncer-tainty about the number of early consumers among the bank’s depositors.For example, different banks could be located in regions subject to indepen-dent liquidity shocks. Third, there is aggregate uncertainty about the fractionof early consumers in the economy. To begin with, we ignore the banks’idiosyncratic uncertainty and focus on individual idiosyncratic uncertaintyand aggregate uncertainty. Aggregate uncertainty is represented by a state ofnature s that takes on two values, H and L, with probability π and 1 − π

respectively. The probability of being an early consumer in state s is denoted byλs where

0 < λL ≤ λH < 1.

We adopt the usual “law of large numbers” convention and assume that thefraction of early consumers in state s is identically equal to the probabilityλs . Note that there is aggregate intrinsic uncertainty only if λL < λH . IfλL = λH = λ then there is no aggregate (intrinsic) uncertainty and the stateof nature s represents extrinsic uncertainty.

All uncertainty is resolved at date 1. The true state s is publicly observedand each consumer learns his type, that is, whether he is an early or a lateconsumer. An individual’s type, early or late, is private information that onlythe individual can observe directly.


There are no asset markets for hedging against aggregate uncertainty atdate 0; for example, there are no securities that are contingent on the state ofnature s. At date 1, there is a market in which future (date-2) consumptioncan be exchanged for present (date-1) consumption. If ps denotes the price ofdate 2 consumption in terms of present consumption at date 1, then one unitof the long asset is worth Ps = psR at date 1 in state s.

Markets are incomplete at date 0, because of the inability to hedge uncer-tainty about the state s. At date 1 markets are complete because all uncertaintyhas been resolved.

We assume that market participation is incomplete: financial institutionssuch as banks can participate in the asset market at date 1, but individualconsumers cannot. Banks are financial institutions that provide investmentand liquidity services to consumers. They do this by pooling the consumers’resources, investing them in a portfolio of short- and long-term assets, andoffering consumers future consumption streams with a better combinationof asset returns and liquidity than individual consumers could achieve bythemselves.

Banks compete by offering deposit contracts to consumers in exchange fortheir endowments and consumers respond by choosing the most attractive ofthe contracts offered. Free entry ensures that banks earn zero profits in equilib-rium. The deposit contracts offered in equilibriummustmaximize consumers’welfare subject to the zero-profit constraint. Otherwise, a bank could enter andmake a positive profit by offering a more attractive contract.

Anything a consumer cando, the bank cando. So there is no loss of generalityin assuming that consumers deposit their entire endowment in a bank atdate 0. Consumers cannot diversify by spreading their money across morethan one bank. The bank invests y units per capita in the short asset and 1 − yunits per capita in the long asset and offers each consumer a deposit contract,which allows the consumer to withdraw either d1 units at date 1 or d2 units atdate 2.Without loss of generality, we set d2 = ∞. This ensures that consumersreceive the residue of the bank’s assets at date 2. Without this assumption thedepositors’ expected utility would not be maximized. The deposit contract ischaracterized by the promised payment at date 1. In what follows we write dinstead of d1.

If ps denotes the price of future consumption at date 1 in state θ , thenthe value of the bank’s assets at date 1 is y + psR(1 − y). Note that, unlike theDiamond–Dybvigmodel, the value of these assets does not depend onwhetherthe bank is in default or not. Because there is a competitive market on whichassets can be sold, the bank’s portfolio is always marked-to-market.

Inwhat follows,we assume that bank runs occur only if they are unavoidable.In other words, late consumers will withdraw at date 2 as long as the bank can

5.1 Markets, Banks, and Consumers 131

offer them as much at date 2 as the early consumers receive at date 1. In theevent of bankruptcy, the bank is required to liquidate its assets in an attemptto provide the promised amount d to the consumers who withdraw at date 1.If the bank does not have enough resources to pay all of the early withdrawersan amount d , then there will be nothing left at date 2 for late consumers whodo not join the run. Hence, in equilibrium, all consumers must withdraw atdate 1 and each consumer will receive the liquidated value of the portfolioy + psR(1 − y).

Under what conditions will a bank be forced to default on its date 1 promisesand liquidate its assets? At date 1, all uncertainty is resolved. People find outtheir individual types and the aggregate state s is realized. Early consumerswill withdraw their deposits from the bank for sure. Late consumers have theoption of leaving their money in the bank, but they could also pretend to beearly consumers, withdraw their funds at date 1, and carry them forward todate 2 using the short asset. Late consumers will be willing to wait until date 2to withdraw if they are confident that they will receive at least d units of thegood at date 2. Otherwise, they will run on the bank and withdraw d at date 1.The cheapest way to satisfy the late consumers, so that they wait until date 2 towithdraw, is to give them d at date 2. If the late consumers receive exactly d atdate 2, the present value of the claims on the bank is λd + (1 − λ)psd . If

λd + (1 − λ)psd ≤ y + psR(1 − y), (5.1)

it is possible to pay the late consumers at least d so that they will be willingto wait until date 2 to withdraw. Otherwise, the bank cannot possibly honorits promise to give each early consumer d and, at the same time, give lateconsumers at least d . A run on the bank at date 1 is inevitable.

If condition (5.1) is satisfied, the deposit contract is said to be incentive-compatible, in the sense that it is optimal for the late consumers to withdrawat date 2. We often refer to the inequality in (5.1) as the incentive constraint,although it also assumes that the bank’s budget constraint is satisfied.

If the bank chooses (d , y) at date 0, the depositor’s consumption at date t instate s is denoted by cts(d , y) and defined by

c1s(d , y) ={

d if (5.1) is satisfied

y + psR(1 − y) otherwise,

c2s(d , y) =

y + psR(1 − y) − λsd

(1 − λs)psif (5.1) is satisfied

y + psR(1 − y) otherwise.


If (5.1) is satisfied then c1s is simply the promised amount d . Understandingc2s is a little more complex. The present value of the firm’s assets at date 1 isy + psR(1 − y). Of this, λsd is paid out to early consumers so the remaindery + psR(1 − y) − λsd is available to pay for the late consumers’ consumption.Since the price of future consumption is ps , we divide by ps to find the totalamount of consumption available at date 2. There are 1 − λs late consumers,so we have to divide by 1 − λs to find the consumption of an individual lateconsumer. Thus, each late consumer receives [y+psR(1−y)−λsd]/[(1−λs)ps]at date 2.

If (5.1) is not satisfied, then there is a crisis and all consumers, both earlyand late, try to withdraw. The bank goes bankrupt and is liquidated fory + psR(1 − y). The early consumers consume their share at date 1 whilethe late consumers carry over their share to date 2 using the short asset. Sincethey do not have access to the asset market, we do not divide by ps .

Using this notation, the bank’s decision problem can be written as

max E[λU (c1s) + (1 − λ)U (c2s)]s.t. 0 ≤ d , 0 ≤ y ≤ 1.

(5.2)

A bank’s choice of deposit contract and portfolio (d , y) is optimal for the givenprice vector p = (pH , pL

)if it solves (5.2).

As we saw in the preceding chapter, the asset market at date 1 can only clearif the price of future consumption is less than or equal to 1. If ps > 1, the assetprice is Ps = psR > R and the banks would only be willing to hold the shortasset. Then the market for the long asset cannot clear. If ps = 1 then bankswill be indifferent between holding the short asset and the long asset betweendates 1 and 2, since in both cases the return on one unit of the good investedat date 1 is one unit of the good at date 2. On the other hand, if ps < 1 thenno one is willing to invest in the short asset at date 1 and everything is investedin the long asset. This is consistent with market clearing, because there is nostock of the short asset unless someone chooses to invest in it at date 1.

Proposition 1 For any state s, the asset market at date 1 clears only if ps ≤ 1.If ps = 1, banks are willing to hold both assets at date 1. If ps < 1, then onlythe long asset is held by banks at date 1.

5.2 TYPES OF EQUILIBRIUM

In order to understand the different forms that equilibrium can take, it ishelpful to consider some simple examples.

5.2 Types of Equilibrium 133

Example 1

U (c) = ln(c);

R = 1.5;

λs ={

0.8 if s = L0.8 + ε if s = H ;

(π , 1 − π) = (0.35, 0.65) .

5.2.1 Fundamental equilibrium with no aggregate uncertainty

We start by considering the case where ε = 0, so λH = λL = 0.8 and s rep-resents extrinsic uncertainty, and look at the fundamental equilibrium, wheres plays no role. If there is no uncertainty, the bank can promise the deposit-ors any consumption allocation that satisfies the budget constraint and theincentive constraint by putting the early consumers’ promised consumptionc1 equal to d and putting the late consumers’ consumption c2 equal to theresidual value of the portfolio. Hence, in the fundamental equilibrium we donot need to concern ourselves with the form of the deposit contract. We canassume, without essential loss of generality, that the bank offers the depositorsa consumption bundle (c1, c2).

Since there is no aggregate uncertainty, banks use the short asset to provideconsumption for the early consumers at date 1 and will use the long asset toprovide consumption for the late consumers at date 2. Since the fraction ofearly consumers is 0.8, the amount of the short asset in the portfolio mustsatisfy (0.8) c1 = y or c1 = y/ (0.8). Similarly, the amount of the long assetmust satisfy (0.2) c2 = R(1 − y) or c2 = R (1 − y) / (0.2), where 1 − y is theamount invested in the long asset and R = 1.5 is the return on the long asset.The banks’ decision problem then is

max 0.8U (c1) + 0.2U (c2)

s.t. 0 ≤ y ≤ 1

c1 = y/ (0.8)

c2 = R(1 − y)/ (0.2) .

Substituting the expressions for c1 and c2 from the budget constraints into theobjective function and using U (c) = ln(c), the problem reduces to

max 0.8 ln( y0.8

)+ 0.2 ln

(R(1 − y)

0.2

).


Assuming an interior solution with regard to y , the necessary and sufficientfirst-order condition for a solution to this problem is

0.8

y= 0.2

1 − y ,

which can be solved for

y = 0.8.

It follows that,

c1 = y

0.8= 1;

c2 = 1.5(1 − y)

0.2= 1.5;

and

E [u(c1, c2)] = 0.8 ln(1) + 0.2 ln(1.5) = 0.081.

As we have seen before, when there is no uncertainty about the future,the asset markets at date 0 can only clear if the asset price at date 1 satisfiesPH = PL = 1. This is the only price at which banks are willing to hold bothassets at date 0 because it is the only price that equalizes the holding returns ofthe two assets between date 0 and date 1.

In each state s, the asset price Ps = 1 implies that one unit invested in thelong asset at date 1 has a return of R = 1.5. Thus, the return on the long assetdominates the return on the short asset between date 1 and date 2 and bankswill only hold the long asset at date 1.

The equilibrium that we have described for Example 1 is autarkic. Each bankcan provide the required consumption for its depositors at each date withoutmaking use of the asset market at date 1. This is not the only fundamentalequilibrium, in fact there aremany,but they differ only in the investmentsmadeby banks and not with respect to the aggregate investment, consumption, andexpected utility of the depositors. We have argued that banks are indifferentbetween holding the two assets at date 0, which yield the same return at date 1,in any fundamental equilibrium. So any portfolio is optimal for them. As longas the aggregate or average portfolio satisfies y = 0.8, there is no reason forbanks not to hold different individual portfolios. For example, suppose that afraction 0.8 of the banks hold only the short asset and a fraction 0.2 hold onlythe long asset. At date 1, a bank holding only the short asset will use 80% of itsportfolio satisfying the early consumers who want to withdraw and with the


remainder will buy 0.2 units of the long asset to provide the late consumerswith consumption of 1.5 at date 2. Banks that have only held the long asset willliquidate 80% of their portfolio in order to obtain consumption for their earlyconsumers and will hold onto the remaining 20% to provide for consumptionof the late consumers at date 2.

Although the asset market is used in this example, it is not essential. Abank can still achieve the first best without using the market. If banks receiveidiosyncratic liquidity shocks,on theother hand, theywill havedifferent liquid-ity needs at date 1. Some banks will have surplus liquidity and other banks willhave a liquidity deficit. These surpluses and deficits can only be removed byusing the market. At the end of this chapter, we will consider a case where theasset market plays an essential role, but for the moment we continue to ignoreidiosyncratic shocks.

5.2.2 Aggregate uncertainty

Now suppose that ε = 0.01, so that λL = 0.8 and λH = 0.81. Uncertaintyabout the state s gives rise to aggregate intrinsic uncertainty. Although theamount of aggregate uncertainty is small, the equilibrium we find for theeconomy with aggregate uncertainty is quite different from the fundamentalequilibrium discussed above. We can easily identify some reasons for the dra-matic effect of a small degree of aggregate uncertainty in the inelastic supplyof and demand for liquidity. Suppose that all banks make identical choicesat date 0. We shall later see that this assumption is not plausible, but it willserve to illustrate some important points. If the investment in the short assetis y units, then the total amount of the good available for consumption in theeconomy at date 1 is y units per capita, independently of the aggregate stateand the prevailing asset prices. In this sense, the aggregate supply of liquidityis inelastic. We denote the aggregate supply of liquidity at date 1 by S (P) anddefine it by putting

Ss(P) = y

for all price levels P and states s.Suppose that each bank also chooses a deposit contract that promises d units

of the good to any depositor who withdraws at date 1. Assuming there are nobank runs and, hence, no defaults, the only demand for liquidity at date 1 is toprovide goods to the early consumers. Since the fraction of early consumers is0.8 in state s = L and 0.81 in state s = H , the aggregate amount of the goodneeded by banks is (0.8) d with probability 0.65 and (0.81) d with probability


0.35. LetDs (P) denote the aggregate demand for liquidity when the asset priceis P and the state is s. Then

Ds(P) ={

(0.8) d if s = L,(0.81) d if s = H .

The demand is price inelastic, because the promised payment d is inelastic, butthe aggregate demand varies with the state.

Market clearing requires that the demand for liquidity does not exceed thesupply, that is,

Ds(Ps) ≤ Ss(Ps)for each state s = H , L. If the market clears in the high state s = H , however,there will be an excess of liquidity in the low state s = L, because

(0.8) d < (0.81) d ≤ y .As we have seen in our earlier discussions of equilibrium in the asset market,when there is excess liquidity at date 1 the price of the long asset will rise toPL = R = 1.5. Only at this price will the returns on the short and long assetbe equalized, so that banks are willing to hold the short asset until date 2.But banks must also hold both assets at date 0 and they will not do that ifthe short asset is dominated by the long asset. If PL = 1.5 the short asset isundominated only if the asset price is lower in the high state. In fact, we musthave PH 1. So the inelasticity of demand and supply means that even smallaggregate liquidity shocks will cause large fluctuations in asset prices.

We have shown that, even in the absence of default, the equilibrium with asmall amount of aggregate uncertainty ε = 0.01 will be quite different fromthe fundamental equilibrium in the limit economy where ε = 0. If we allowfor the possibility of default, we discover another difference: banks that startout looking very similar will adopt quite different investment and risk-sharingstrategies. To see this, suppose to the contrary that all the banks continue tomake the same choices at the first date. As we assumed above, each bank investsy units of the good in the short asset and offers a deposit contract promisingd units of consumption to depositors who withdraw at date 1.We have arguedthat a bank will default if and only if it fails to satisfy the incentive constraint(5.1) at date 1. But if every bank has made the same choices at date 1, eitherthey will all violate the incentive constraint or none will. If every bank defaults,there will be no one to buy the long assets that must be liquidated at date 1and the asset price will fall to zero. This cannot be an equilibrium, however. Ata price of zero, someone would be tempted to deviate, choose y and d so that


bankruptcy is avoided, and make a large capital gain by purchasing the longassets when their price falls to zero. So, in order to have default in equilibrium,equilibriumwill have to bemixed in the sense that there are two types of banks,call them safe banks and risky banks, that make very different choices at date 0.

Safe banks hold a lot of the short asset and offer deposit contracts promisinglowpayments at date 1. Risky banks hold a lot of the long asset and offer depositcontracts promising high payments at date 1.When liquidity demands are low(s = L) the safe banks have excess liquidity which they supply to the market bybuying the long asset. Risky banks obtain the liquidity they need to honor theirdeposit contracts by selling the long asset. When liquidity demands are high(s = H ) the market for the long asset is less liquid because the safe banks mustdevote more of their liquidity to satisfying the needs of their own customers.This liquidity shortage leads to a drop in the price of the long asset, whichforces the risky banks to go bankrupt and liquidate their stocks of the longasset. The increase in the supply of the long asset can lead to a sharp drop inprices. In this case there is “cash-in-the-market” pricing. The safe banks holdjust enough liquidity in excess of their customers’ needs to enable them to buyup the long asset at a firesale price. The low price compensates them for thecost of holding the extra liquidity when liquidity demands are low and pricesare high.

To see how such an equilibrium operates in detail, consider the followingexample in which the parameter values are the same as in Example 1 exceptthat now ε = 0.01 rather than ε = 0. This variant with aggregate uncertaintyis referred to as Example 1A.

Example 1A

U (c) = ln(c);

R = 1.5;

α = 0.8;

π = 0.35;

ε = 0.01.

Then

λs ={

0.8 with probability 0.65;

0.81 with probability 0.35.


The asset price Ps is random and takes the values

Ps ={pHR = (0.430) (1.5) = 0.645 if s = H ;

pLR = (0.940) (1.5) = 1.410 if s = L.Next we describe the behavior of the safe and risky banks. A proportion

ρ = 0.979 of the banks adopt the safe strategy of avoiding bankruptcy anddefault. Specifically, they choose to invest a large amount yS in the short assetand promise a low amount dS to depositors who withdraw early.

yS = 0.822; dS = 0.998.

Both choices make it easier for the bank to satisfy the incentive constraint.Once these choices are made, the prices and budget constraints determine thedepositors’ consumption at each date and in each state:

cSs ={ (cS1H , cS2H

) = (0.998, 1.572) if s = H ;(cS1L , c

S2L

) = (0.998, 1.461) if s = L.Note that early consumers receive the same level of consumption (dS) in eachstate. The late consumers have different levels of consumption in each statebecause they receive the residual value of the portfolio and this depends onthe asset price Ps . Note that the late consumers are better off in the high state:the low asset price allows the bank to purchase future consumption cheaply.Alternatively, one can think of the bank as making capital gains from buyingup long assets cheaply at date 1 and selling them more dearly at date 2.

A proportion 1 − ρ = 0.021 of the banks adopt the risky strategy that willlead to default and bankruptcy when prices fall in state s = H . Unlike the safebanks, they choose a low level of yR and promise a high level of consumptionto early withdrawers.

yR = 0.016; dR = 1.405.

Both of these choices make it harder to satisfy the incentive constraint.The consumption levels determined by these choices and by the budgetconstraints are

cRs ={ (cR1H , cR2H

) = (0.651, 0.651) if s = H ;(cR1L , c

R2L

) = (1.405, 1.486) if s = L.In the low state, the bank is solvent and can give both early and late consumerswhat was promised. In the high state, by contrast, the bank goes bankrupt


and defaults on its promises and the early and late consumers both receive theliquidated value of the bank’s portfolio.

In equilibrium, individuals must be indifferent between depositing theirmoney in a safe or risky bank. Otherwise, one type of bank will attract nocustomers. If we compute the expected utility of consumption for depositorsin the safe and risky banks, we find that both are equal to 0.078.

To check that these choices are consistent with equilibrium, the first thingwe need to show is that markets clear at date 1. Consider what happens in states = L. The safe banks can satisfy the liquidity demand of their depositors fromtheir own holdings of the safe asset. A fraction λL = 0.8 of their customers areearly consumers, so the demand for liquidity is λLdS = 0.8 × 0.998 = 0.798.Then the safe banks will each have yS − λLdS = 0.822 − 0.798 = 0.024 unitsof the good left over. Since the proportion of safe banks is ρ = 0.979, the totalamount of excess liquidity is ρ(yS − λLdS) = 0.979 × 0.024 = 0.023. SincePL = 0.940 < 1 the short asset is dominated between date 1 and date 2, sothe safe banks will only want to hold the long asset. Accordingly, they supplythe entire 0.023 units of the good in exchange for the long asset.

Next consider the risky banks. Since they hold none of the short asset,they must sell part of their holdings of the long asset in order to provide thepromised liquidity to their early consumers. The demand for liquidity fromtheir customers is λLdR = 0.8 × 1.405 = 1.124. They have 0.016 from theirholding of the short asset so they need 1.124 − 0.016 = 1.108. Since theproportion of risky banks is 1 − ρ = 0.021 the total demand for liquidity is(1 − ρ) λLdR = 0.021 × 1.108 = 0.023. Thus demand for liquidity equalssupply when the state s = L.

In state s = H , the safe banks’ supply of liquidity can be calculated in thesame way, simply taking note that the proportion of early consumers is nowλH = 0.81. The supply of liquidity is

ρ(yS − λHd

S) = (0.979) (0.822 − 0.81 × 0.998)

= 0.013.

The risky banks have (1−yR) = 0.987 units of the long asset. Because they arebankrupt, they must liquidate all their assets and this means they must supplythe whole amount of the long asset at date 1. Implicitly, they are demandingPH = 0.645 units of liquidity in exchange. Since there are (1 − ρ) = 0.021risky banks, the total demand for liquidity is

(1 − ρ) PH(1 − yR) = 0.021 × 0.645 × 0.987 = 0.013.

Thus, the asset market clears in state s = H too.


To show thatmarkets clear at date 0, it is necessary to show that the portfolioheld by each bank is optimal. This requires us to check the first-order condi-tions that show the effect of a small change in the portfolio on consumptionin each state and on expected utility. This is a rather complicated exercise thatwe will not pursue here.

The example has shown that a small increase in ε = λH − λL leads to verydifferent price behavior compared to the fundamental equilibrium. It alsoleads to the possibility of banks defaulting in equilibrium. As we have arguedabove, substantial asset price volatility is a general property of equilibria withintrinsic uncertainty, however small that uncertainty may be. This leads usto ask two questions. First, what sort of equilibrium would we observe ifwe let ε > 0 converge to zero? Second, would the limit of this sequence ofequilibria be an equilibrium of the limit economy where ε = 0? The answerto both questions is that the equilibrium with intrinsic uncertainty convergesto a sunspot equilibrium of the limit economy. In this sunspot equilibrium,prices fluctuate even though there is no intrinsic aggregate uncertainty in theeconomy.

5.2.3 Sunspot equilibria

To describe the limit of the equilibria with intrinsic uncertainty, we return toExample 1. The parameters are identical but another equilibrium is found. Todistinguish this case we refer to it as Example 1S.

Example 1S The asset price is random

Ps ={pHR = (0.432) (1.5) = 0.648 if s = H ;

pLR = (0.943) (1.5) = 1.415 if s = L.

The proportion of safe banks is ρ = 1 and their choices are

yS = 0.8; dS = 1.

These choices and the equilibrium prices determine the consumption in eachstate at each date:

cSs ={ (cS1H , cS2H

) = (1.0, 1.5) if s = H ;(cS1L , c

S2L

) = (1.0, 1.5) if s = L. (5.3)


Note that consumption is exactly the same as in the fundamental equilibrium.The expected utility obtained from this consumption is, as we claimed earlier,0.081.

Risky banks on the other hand, continue to choose a low investment in theshort asset yR and promise high consumption to early withdrawers dR :

yR = 0; dR = 1.414

and the corresponding consumption is

cRs ={ (cR1H , cR2H

) = (0.648, 0.648) if s = H ;(cR1L , c

R2L

) = (1.414, 1.500) if s = L.

Now the risky banks invest nothing in the short asset and enter bankruptcyin the high state s = H . If we were to calculate the expected utility of thisconsumption plan, we would find that it equals the expected utility of the safebanks, 0.081. Thus, consumers are indifferent between the two types of banks,even though in equilibrium there are no risky banks since ρ = 1.

It may seem odd that it is optimal for banks to adopt a risky strategy eventhough none attempt to do so. In fact, this is a necessary requirement for anyequilibrium that is the limit of equilibria as ε converges to zero from above. Inthe equilibria with ε > 0, a positive fraction of the banks were risky, implyingthat it must be optimal for risky banks to operate. As ε becomes vanishinglysmall, so does the fraction of risky banks, but it remains optimal in the limitfor risky banks to operate, even though none choose to do so. The absenceof default in the limit equilibrium results from the fact that, as we saw inthe fundamental equilibrium, banks can achieve the first best without usingthe asset market. The only way for everyone to get the first best expectedutility is for all banks to choose the safe strategy. Even though it is optimalfor a single bank (of negligible size) to adopt the risky strategy, if a positivemeasure of banks were to do so, the depositors’ welfare would be reducedbelow the first-best level. Thus, in the limit economy with ε = 0, equilibriumrequires ρ = 1.

At the allocation given in (5.3), depositors bear no risk and so are approxi-mately risk neutral in the face of small risks. They will be indifferent betweenholding a bit more or less of each asset if and only if the following condition issatisfied:

πR

PH+ (1 − π)

R

PL= R. (5.4)


The right hand size is just the expected return at date 2 to an investment ofone unit in the long asset at date 0. The left hand side is the expected return tothe following strategy: invest one unit in the short asset at date 0 and use thereturns at date 1 to buy as much of the long asset as possible. With probabilityπ the high state occurs, the price is PH , and the strategy results in the purchaseof 1/PH units of the long asset; with probability 1−π the low state occurs, theprice is PL , and one obtains 1/PL units of the long asset. Each unit of the longasset yields R at date 2 so the expected return from the investment strategy isequal to the left hand side of (5.4). In other words, the expected return fromholding the short asset is the same as the expected return from holding thelong asset if and only if (5.4) holds. It can easily be checked that

0.65 × 1.5

1.415+ 0.35 × 1.5

0.648= 1.5,

so the asset market does clear at date 1 in the limiting sunspot equilibrium.There are many other sunspot equilibria. In fact, any pair of prices between

0 and 1.5 corresponds to a sunspot equilibrium if it satisfies (5.4). Such a largeset of price vectors can clear the market and support an equilibrium preciselybecause the market is not needed when there is no aggregate uncertainty. Theonly function of the prices is to make sure (a) that banks are willing to holdthe appropriate amounts of both assets at date 0 and (b) that none of themwant to use the asset market. Condition (a) is guaranteed by the first-ordercondition (5.4) and condition (b) is guaranteed by the fact that banks havejust enough liquidity to pay off their early withdrawers at date 1 and so havenothing to trade at date 1. The fact that Ps < R for each state s implies thatbanks only want to hold the long asset between dates 1 and 2 anyway.

The solutions to the variants of Example 1 are shown in Table 5.1. Panel1F shows the fundamental equilibrium, Panel 1A shows the equilibrium withaggregate uncertainty, and Panel 1S shows the sunspot equilibrium.

5.2.4 Idiosyncratic liquidity shocks for banks

So farwe have assumed that banks do not receive idiosyncratic liquidity shocks,that is, all banks have the sameproportion of early and late consumers. Supposenext that banks can receive different proportions of early consumers. In thiscase all banks must use the markets to trade. If a bank has a high liquidityshock it needs to acquire liquidity; if it has a low liquidity shock it supplies


Table 5.1. Equilibria for Example 1.

� ε

[E[US]E[UR]

] [yS

yR

]

(cS1H , cS2H

)(cS1L , c

S2L

)(cR1H , cR2H

)(cR1L , c

R2L

)

[pHpL

]

1F 0 0.081 0.800 (1.000, 1.500) 0.667

1A 0.010

[0.0780.078

] [0.8220.016

]

(0.998, 1.572)(0.998, 1.461)(0.651, 0.651)(1.405, 1.486)

[

0.4300.940

]

1S 0

[0.0810.081

] [0.800

0

]

(1.000, 1.500)(1.000, 1.500)(0.648, 0.648)(1.414, 1.500)

[

0.4320.943

]


� ε

[E[US]E[UR]

] [yS

yR

]

(cS1H , cS2HL , c

S2HH

)(cS1L , c

S2LL , c

S2LH

)(cR1H , cR2HL , c

R2HH

)(cR1L , c

R2LL , c

R2LH

)

[pHpL

]

2F 0 0.081 0.800 (1.000, 1.500) 0.667

2A 0.010

[0.0770.077

] [0.814

0

]

(0.995, 1.715, 1.392)(0.995, 1.415, 1.627)(0.678, 0.678, 0.678)(1.360, 1.502, 1.503)

[

0.4520.907

]

2S 0

[0.0800.080

] [0.798

0

]

(0.998, 1.648, 1.281)(0.998, 1.430, 1.652)(0.681, 0.681, 0.681)(1.364, 1.502, 1.503)

[

0.4540.910

]

liquidity to the market. The main effect of the idiosyncratic shocks is that theylower the expected utility by forcing banks to trade at volatile prices.

To see the effect of introducing idiosyncratic liquidity shocks for banks weconsider the following a variation on Example 1. Instead of assuming that λs is


purely a function of the state of nature, we assume that it is a random variableλs in each state s.

Example 2

U (ct ) = ln(ct );

R = 1.5;

λH ={

0.75 + ε w. pr. 0.5;0.85 + ε w. pr. 0.5.

λL ={

0.75 w. pr. 0.5;0.85 w. pr. 0.5.

π = 0.35.

The equilibria of this example are shown in Table 5.2. The fundamentalequilibrium shown in Panel 2F is unchanged from Panel 1F in Table 5.1. SincePH = PL = 1 it is possible to trade the long asset at date 1 and acquirethe needed liquidity without any effect on depositor welfare. The sunspotequilibria in Panel 1S in Table 5.1 and Panel 2S in Table 5.2 are different,however. Trade occurs at the low price and this adversely affects the welfare ofthe safe banks as well as the risky banks. To mitigate the effects of trading atthis low price, the safe banks lower dS , from 1 in Panel 1S to 0.995 in Panel2S. Similarly, yS is reduced to 0.798 from 0.800. Expected utility is loweredfrom 0.081 in Panel 1S to 0.080 in Panel 2S. As in the autarkic equilibria withno idiosyncratic risk, introducing a small amount of aggregate uncertaintyeliminates equilibriawith a non-stochastic price.Only the sunspot equilibriumis robust. Panel 2A shows the equilibrium with intrinsic uncertainty (ε =0.010) is very close to the sunspot equilibrium. In this case ρ = 0.989 and1 − ρ = 0.011, so there is entry by both safe and risky banks.

5.2.5 Equilibrium without bankruptcy

In the examples considered above, bankruptcy is always optimal, even if itwas not always observed in equilibrium. There are also equilibria in whichbankruptcy is not optimal. In other words, we have a financial crisis thatinvolves extreme price volatility but no bankruptcy. We next consider anexample of this type. For simplicity, we return to the case where there is noidiosyncratic risk for banks.



� ε

[E[US]E[UR]

] [yS

yR

]

(cS2H , cS2H

)(cS2L , c

S2L

)(cR2H , cR2H

)(cR2L , c

R2L

)

[pHpL

]

3F 0 0.203 0.500 (1.000, 1.500) 0.667

3A 0.010

[0.1990.110

] [0.508

0

]

(0.996, 1.507)(0.996, 1.497)(0.561, 0.561)(1.500, 1.500)

[

0.3741

]

3S 0

[0.2030.111

] [0.500

0

]

(1.000, 1.500)(1.000, 1.500)(0.563, 0.563)(1.500, 1.500)

[

0.3751

]

Example 3

U (c) = ln(c);

R = 1.5;

λs ={

0.5 if s = L;0.5 + ε if s = H .

π = 0.3.

The equilibria of this example are shown in Table 5.3. The fundamentalequilibrium, where ε = 0, is shown in Panel 3F. It can straightforwardly beshown, as in Examples 1 and 2, that c1 = 1, c2 = R = 1.5 and pH = pL =0.667, so PH = PL = 1. The expected utility is 0.5 ln(1) + 0.5 ln(1.5) = 0.203.

Next consider the equilibrium with aggregate uncertainty, where ε = 0.01.This is shown in Panel 3A. The key feature of this equilibrium is that only thesafe banks enter. Their holdings of the short asset, yS = 0.508, are enough tocover their liquidity needs when s = H , i.e. liquidity demand is high. The safebanks need λHdS = 0.51 × 0.996 = 0.508 units of the good in the high state.This means that there is excess liquidity in the low state s = L. As we have seen,the only possible equilibrium price is pL = 1, so PL = pLR = 1.5. Given thisprice in state s = L it follows that pH 1 is necessary in order for the banks tohold both the short and long assets between dates 0 and 1 (see Section 5.2.2).


In this example, pH = 0.374. The safe banks offer the consumption plan

cSs ={ (cS1H , cS2H

) = (0.996, 1.507) if s = H ;(cS1L , c

S2L

) = (0.996, 1.497) if s = L.The expected utility provided to depositors by the safe banks is 0.199.

If a risky bank were to enter, the best that it could do is to choose

yR = 0; dR = 1.5

and offer depositors the consumption plan

cRs ={ (cR1H , cR2H

) = (0.561, 0.561) if s = H ;(cR1L , c

R2L

) = (1.500, 1.500) if s = L.The expected utility a depositor would receive from the contract offered by therisky banks is 0.110, which is strictly worse than that offered by the safe banks.Thus, there will be no risky banks operating in equilibrium.

Panel 3S shows the sunspot equilibrium which is the limit of fundamen-tal equilibria for economies with aggregate intrinsic uncertainty as ε ↘ 0.The allocation provided by the safe bank is the same as in the fundamentalequilibrium and so is the expected utility, 0.203. We have pL = 1, as in thecase with aggregate uncertainty. Here, using equation (5.4), we can show thatpH = 0.375. At these prices, the best a risky bank can do is provide customerswith expected utility 0.111, so again there is no entry.

Note that whether we have bankruptcy or not, there is price volatility inevery equilibrium except the fundamental equilibrium.

5.2.6 Complete versus incomplete markets

The key feature of themarket structure in this chapter is that there are noArrowsecurities so markets are incomplete. When these securities are introduced theanalysis is changed significantly. Rather than the sunspot equilibrium con-sidered in the examples being the robust equilibrium, it is the fundamentalequilibrium that becomes robust. This now becomes the limit equilibrium asε ↘ 0. For small ε the equilibria are very close to the fundamental equilib-rium. When ε = 0 the sunspot equilibria are ruled out because of the tradingin Arrow securities. The crucial determinant of the form of equilibrium isthus whether markets are complete or incomplete. We consider the completemarkets case in the next chapter.

5.3 Relation to the Literature 147

5.3 RELATION TO THE LITERATURE

The model presented in this chapter is related to the wider literature onsunspots and general equilibrium with incomplete (GEI) markets. The the-oretical analysis of sunspot equilibria beganwith the seminal work of Azariadis(1981) and Cass and Shell (1983), which gave rise to two streams of litera-ture. The Cass–Shell paper is most closely related to work in a Walrasian,general equilibrium framework; the Azariadis paper is most closely related tothe macroeconomic dynamics literature. For a useful survey of applications inmacroeconomics, see Farmer (1999); for an example of the current literaturein the general equilibrium framework, see Gottardi and Kajii (1995, 1999).

It can be shown that sunspots do not matter when markets are complete(for a precise statement, see Shell and Goenka 1997). The incompletenessin our model reveals itself in two ways. First, sunspots are assumed to benoncontractible – that is, the deposit contract is not explicitly contingent onthe sunspot variable. In this respect the model simply follows the incom-plete contracts literature (see, e.g., Hart 1995). Second, there are no marketsfor Arrow securities contingent on the sunspot variable, so financial institu-tions cannot insure themselves against asset-price fluctuations associated withthe sunspot variable. This is the standard assumption of the GEI literature(see, e.g., Geanakoplos 1990 or Magill and Quinzii 1996).

The results of this chapter help us understand the relationship betweentwo traditional views of financial crises discussed in Chapter 3. One view isthat they are spontaneous events, unrelated to changes in the real economy.Historically, banking panics were attributed to“mob psychology”or“mass hys-teria” (see, e.g. Kindleberger 1978). Themodern version of this theory explainsbanking panics as equilibrium coordination failures (Bryant 1980; Diamondand Dybvig 1983). An alternative view is that financial crises are a naturaloutgrowth of the business cycle (Gorton 1988; Calomiris and Gorton 1991;Calomiris and Mason 2000; Allen and Gale 1998, 2000a–c). The model of thischapter combines the most attractive features of both traditional approaches.Like the sunspot approach, it produces large effects from small shocks. Like thereal business cycle approach, it makes a firm prediction about the conditionsunder which crises will occur.

There is a small but growing literature related to financial fragility in thesense that the term is used in this chapter. Financialmultiplierswere introducedby Bernanke and Gertler (1989). In the model of Kiyotaki and Moore (1997),the impact of illiquidity at one link in the credit chain has a large impactfurther down the chain. Chari and Kehoe (2000) show that herding behaviorcan cause a small information shock to have a large effect on capital flows.


Lagunoff and Schreft (2001) show how overlapping claims on firms can causesmall shocks to lead to widespread bankruptcy. Bernardo and Welch (2004)develop a model of runs on financial markets and asset-price collapses basedon the anticipation of liquidity needs.

Postlewaite et al. (2003) study amodel inwhich transactions in a competitivemarket are preceded by fixed investments. They show that, in the absence offorward contracts, equilibrium spot prices are highly volatile even when thedegree of uncertainty is very small.

5.4 DISCUSSION

At the beginning of the chapter we stressed four points. The first was that smallshocks could have large effects because of the interaction of banks andmarkets.We have investigated endogenous crises, where small or negligible shocks set offself-reinforcing and self-amplifyingprice changes. In the limit,when the shocksbecome vanishingly small, there is no aggregate exogenous uncertainty, butthis does not mean that there is no endogenous uncertainty. We distinguishedtwo kinds of uncertainty. Intrinsic uncertainty is caused by stochastic fluc-tuations in the primitives or fundamentals of the economy. Examples wouldbe exogenous shocks that effect liquidity preferences or asset returns. Extrinsicuncertainty by definition has no effect on the fundamentals of the economy.Anequilibrium with no extrinsic uncertainty is called a fundamental equilibrium,because the endogenous variables are functions of the exogenous primitivesor fundamentals of the model (endowments, preferences, technologies). Anequilibrium with extrinsic uncertainty is called a sunspot equilibrium, becauseendogenous variables may be influenced by extraneous variables (sunspots)that have no direct impact on fundamentals. A crisis cannot occur in a funda-mental equilibrium in the absence of exogenous shocks to fundamentals, suchas asset returns or liquidity demands. In a sunspot equilibrium, by contrast,asset prices fluctuate in the absence of aggregate exogenous shocks, and crisesappear to occur spontaneously.

Our second point was the key role of liquidity in determining asset prices.The supply of liquidity is determined by the banks’ initial portfolio choices.Subsequently, small shocks to the demand for liquidity, interacting with thefixed supply, cause a collapse in asset prices. Once the supply of liquidity isfixed by the banks’ portfolio decisions, shocks to the demand for liquidity cancause substantial asset-price volatility and/or default. The supply of liquidity isfixed in the short run by the banks’ portfolio decisions at date 0. In the absenceof default, the demand for liquidity is perfectly inelastic in the short run. Ifthe banks’ supply of liquidity is sufficient to meet the depositors’ demand

5.4 Discussion 149

when liquidity preference is high, there must be an excess supply of liquiditywhen liquidity preference is low. The banks will be willing to hold this excessliquidity between dates 1 and 2 only if the interest rate is zero (the price ofdate-2 consumption in terms of date-1 consumption is 1). A low interest rateimplies that asset prices are correspondingly high.However, asset prices cannotbe high in all states for then the short asset would be dominated at date 0 andno one would be willing to hold it. So, in the absence of default, there willbe substantial price volatility. This argument does not require large shocks toliquidity demand.

Our third point was the role of mixed equilibria, in which ex ante identicalbanks must choose different strategies. For some parameter specifications, weshow that one group of banks follows a risky strategy by investing almost all oftheir funds in the long asset. They meet their demands for liquidity by sellingthe asset in themarket. Another group of banks follows a safe strategy and holda large amount of the short asset. The safe banks provide liquidity to the riskybanks by purchasing the risky banks’ long-term assets. Safe banks also provideliquidity to each other: because there are idiosyncratic shocks to liquiditydemand, the safe banks with a high demand for liquidity sell long-term assetsto those with a low demand.

Finally, our fourth point concerned the difference between systemic riskand economy-wide crises. There are important differences between the presentmodel of systemic or economy-wide crises and models of individual bank runsor panics of Bryant (1980) and Diamond and Dybvig (1983) discussed inChapter 3. In the model of this chapter a crisis is a systemic event. It occursonly if the number of defaulting banks is large enough to affect the equilibriumasset price. In the panic model, by contrast, bank runs are an idiosyncraticphenomenon. Whether a run occurs at a particular bank depends on thedecisions taken by the bank’s depositors, independently of what happens atother banks. It is only by coincidence that runs are experienced by severalbanks at the same time.

Another difference between panics and crises concerns the reasons for thedefault. In the Bryant–Diamond–Dybvig story, bank runs are spontaneousevents that depend on the decisions of late consumers to withdraw early. Giventhat almost all agents withdraw at date 1, early withdrawal is a best responsefor every agent; but if late consumers were to withdraw at date 2, then latewithdrawal is a best response for every late consumer. So there are two “equi-libria” of the coordination game played by agents at date 1, one with a bankrun and one without. This kind of coordination failure plays no role in thepresent model. In fact, coordination failure is explicitly ruled out: a bank runoccurs only if the bank cannot simultaneously satisfy its budget constraintand its incentive constraint. From the point of view of a single, price-taking


bank, default results from an exogenous shock. When bankruptcy does occur,it is the result of low asset prices. Asset prices are endogenous, of course, andthere is a “self-fulfilling” element in the relationship between asset prices andcrises. Banks are forced to default and liquidate assets because asset prices arelow, and asset prices are low as a result of mass bankruptcy and the associatedliquidation of bank assets.

Several features of the model are special and deserve further consider-ation. First, we have noted the importance of inelastic demand for liquidity ingenerating large fluctuations in asset prices from small demand shocks. Theinelasticity of demand follows from two assumptions. The first is the assump-tion that banks use demand deposits, which do not allow the payment at date 1to be contingent on demand (or anything else). The second is the assumptionof Diamond–Dybvig preferences, which rule out intertemporal substitution ofconsumption. We see both the Diamond–Dybvig preferences and the use ofdemand deposits as a counterpart to the empirical fact that financial contractsare typically written in a “hard” way that requires strict performance of pre-cisely defined acts, independently of many apparently relevant contingencies.These hard contracts may be motivated by enforcement and incentive prob-lems, but it would be too difficult to include them explicitly in themodel. Thereseems little doubt that such factors are relevant in real markets and should betaken into account here.

Second, an alternative justification for incomplete contracts is that theyprovide a way of modeling – within the standard, Walrasian, auction-marketframework – some realistic features of alternative market-clearing mech-anisms. In an auction market, prices and quantities adjust simultaneously in atatônnement process until a full equilibrium is achieved. An alternative mech-anism is one in which quantities are chosen before prices are allowed to adjust.An example is the use of market orders in markets for company stocks. In thebanking context, if depositors were required to make a withdrawal decisionbefore the asset price was determined in the interbank market, then the sameinelasticity of demand would be observed even if depositors had preferencesthat allowed for intertemporal substitution. There may be other institutionalstructures that have the qualitative features of the model of this chapter.

Finally, pecuniary externalities “matter” in the model because markets areincomplete: if banks could trade Arrow securities contingent on the states θ ,they would be able to insure themselves against changes in asset values. Witha complete set of Arrow securities, risk sharing must be efficient, so in theabsence of intrinsic uncertainty (ε = 0) the only possible equilibrium is theone we have called the fundamental equilibrium. The equilibrium allocation isincentive-efficient, sunspots have no real impact, and there are no crises. Notethat, although the existence of the markets for Arrow securities has an effect

References 151

(by eliminating the other equilibria), there is no trade inArrow securities in thefundamental equilibrium. When intrinsic uncertainty is introduced (ε > 0)the fundamental equilibrium is now robust. It is the limit equilibriumas ε ↘ 0.We consider the casewith completemarkets and contrast it with the incompletemarkets case in the next chapter.

REFERENCES


Allen, F. and D. Gale (2000a). “Financial Contagion,” Journal of Political Economy 108,1–33.

Allen, F. and D. Gale (2000b). “Optimal Currency Crises,”Carnegie Rochester Series onPublic Policy 53, 177–230.

Allen, F. andD.Gale (2000c).“Bubbles andCrises,”The Economic Journal 110, 236–256.Allen, F. and D. Gale (2004). “Financial Fragility, Liquidity, and Asset Prices,” Journalof the European Economic Association 2, 1015–1048.

Azariadis, C. (1981). “Self-Fulfilling Prophecies,” Journal of Economic Theory 25,380–396.

Bernanke, B. and M. Gertler (1989). “Agency Costs, Net Worth, and BusinessFluctuations,”American Economic Review 79, 14–31.

Bernardo, A. and I. Welch (2004). “Financial Market Runs,” Quarterly Journal ofEconomics 119, 135–158.


Calomiris,C. andG.Gorton (1991).“TheOrigins of BankingPanics,Models,Facts, andBank Regulation.” In Financial Markets and Financial Crises, edited by R. Hubbard.Chicago, IL: University of Chicago Press.

Calomiris, C. and J. Mason (2000). “Causes of U.S. Bank Distress During theDepression.” NBER Working Paper W7919.

Cass, D. and K. Shell (1983). “Do Sunspots Matter?” Journal of Political Economy 91,193–227.

Chari, V. and P. Kehoe (2000). “Financial Crises as Herds.” Federal Reserve Bank ofMinneapolis Working Paper.


Farmer, R. (1999). The Macroeconomics of Self-Fulfilling Prophecies. Cambridge andLondon: MIT Press.

Geanakoplos, J. (1990). “An Introduction to General Equilibrium with IncompleteAsset Markets,” Journal of Mathematical Economics 19, 1–38.



Gottardi, P. and A. Kajii (1995). “Generic Existence of Sunspot Equilibria: The RealAsset Case,” University of Pennsylvania, CARESS Working Paper 95/12.

Gottardi, P. and A. Kajii (1999). “The Structure of Sunspot Equilibria: The Role ofMultiplicity,”Review of Economic Studies 66, 713–732.

Hart, O. (1995). Firms, Contracts and Financial Structure. Oxford: Oxford UniversityPress.

Kindleberger, C. (1978).Manias, Panics, and Crashes: AHistory of Financial Crises. NewYork, NY: Basic Books.

Kiyotaki, N. and J. Moore (1997). “Credit Chains,” Journal of Political Economy 99,220–264.

Lagunoff, R. and S. Schreft (2001). “AModel of Financial Fragility,” Journal of EconomicTheory 99, 220–264.

Magill, M. and M. Quinzii (1996). Theory of Incomplete Markets, Volume I. Cambridgeand London: MIT Press.

Postlewaite,A.,G.Mailath and L. Samuelson (2003).“Sunk Investments Lead toUnpre-dictable Prices.” University of Pennsylvania: http://www.ssc.upenn.edu/˜apostlew/.

Schnabel, I. and H. Shin (2004). “Liquidity and Contagion: The Crisis of 1763,” Journalof the European Economic Association 2, 929–968.

Shell, K. and A. Goenka (1997). “When Sunspots Don’t Matter,” Economic Theory 9,169–178.

http://www.ssc.upenn.edu/%CB%9Capostlew/

6

Intermediation and markets

In the preceding chapter we studied financial fragility from the point of viewof positive economics, that is, trying to understand the factors that give riseto financial crises and the reasons why a financial system might be sensitiveto small shocks. We noted in passing that financial crises might be inefficient,but this was not the focus of our analysis. Now it is time to turn to normativequestions and try to understand why financial crises are a “bad thing.”

To understand why financial crises are a “bad thing,” we begin by asking adifferent question “Under what circumstances are financial crises efficient?”We take this indirect approach for several reasons.

• First, we want to challenge the conventional wisdom that financial crises arealways and everywhere a “bad thing.” It may well be true that financial crisesimpose substantial costs on the economy. Certainly there have been manyhistorical episodes inmany countries that suggest the costs of financial crisescan be very substantial. At the same time, any regulation of the financial sys-tem involves costs. The most important of these costs are the distortionsimposed on the financial system by a regulatory regime that restricts whatfinancial institutions may and may not do. To measure the costs and bene-fits of any policy, we need to have a clear understanding of the conditionsfor efficiency in the financial system, including the conditions for efficientfinancial crises.

• A second reason for studying the conditions under which crises are efficientis that knowledge of these conditions may suggest techniques for managingcrises and reducing their costs. Casual observation suggests that the centralbanking techniques developed over the past two centuries are mainly theresult of trial and error, without much rigorous theory behind them. Thefirst step in developing an optimal financial stability policy is to carry outa thorough welfare analysis of financial crises, including the conditions forefficient financial crises.

• A third reason for adopting the proposed approach is that economists havea well developed set of tools for studying optimal economic systems. So

154 Chapter 6. Intermediation and Markets

it is somewhat easier to address normative questions by characterizing theconditions for efficiency rather than the reverse.

One of the lessons of this chapter is the important role of missing markets.Although it may not have been obvious at the time, the properties of themodel studied inChapter 5 depend crucially on the absence of certain financialmarkets, specifically,markets forArrow securities.1Aswe saw,deposit contractscommit intermediaries to provide each depositor who withdraws at date 1 afixed amount of consumption, independently of the state of nature. If thedemand for liquidity is high, the only way for the intermediary to obtainenough liquidity to meet its commitments is through asset sales. From thepoint of view of the intermediary, selling assets is unfortunate for two reasons.First, the intermediary may be forced to dispose of assets at “fire-sale” prices.Depositors receive lower payouts as a result. Second, if a large number ofintermediaries sell at the same time, the selling pressure will drive prices downfurther, forcing the intermediaries to unload even more assets, and worseningthe crisis. These two effects together explain the inefficiency and severity of thefinancial crisis.

There are two types of incompleteness in the preceding chapter’s analysisof financial fragility. We say that a contract is complete if the outcome is (inprinciple) contingent on all states of nature.Deposit contracts are not completein this sense because the amount of consumption promised to withdrawers atdate 1 is fixed, i.e. not contingent on the state of nature.We say thatmarkets arecomplete if there are markets on which intermediaries can trade Arrow secur-ities for each state of nature. These markets allow intermediaries to purchaseliquidity contingent on the state of nature. If there are no markets for Arrowsecurities, then markets are incomplete. Of the two types of incompleteness, itis the incompleteness of markets that accounts for the inefficiency of financialcrises.

To see the importance of missing markets, suppose that markets for Arrowsecurities were introduced to themodel of financial fragility. If an intermediaryanticipated a shortage of liquidity in a particular state of nature, it would bepossible to purchase Arrow securities that pay off in that state in order toprovide extra liquidity and avoid the need for asset sales. This would cut thelink between the demand for liquidity and the sale of assets. As a result, thepricing of assets would be insulated from liquidity shocks and the destabilizingeffects of incomplete contracts would be reduced if not avoided altogether.

1 An Arrow security is a promise to deliver one unit of account (one “dollar” ) if a specifiedstate of nature occurs and nothing otherwise. The concepts of Arrow securities and completemarkets are reviewed in Chapter 2.

6.1 Complete Markets 155

Another benefit of markets for Arrow securities is improved risk sharing.They allow the intermediary to pay for liquidity in the state where it is neededby selling liquidity in the other state. In effect, the intermediary is transferringwealth from a state where the marginal utility of consumption is low to astate where it is high. This is what efficient risk sharing requires. Asset sales,by contrast, force the intermediary to reduce consumption in the state wheremarginal utility is already high, increasing the variation in consumption acrossstates and resulting in inefficient risk sharing.

In the rest of this chapter, we explore the implications of complete marketsfor the efficiency of the incidence of financial crises. Under certain additionalassumptions, which are analogous to the assumptions of the fundamentaltheorems of welfare economics,we shall show that completemarkets guaranteethe efficiency of laisser-faire equilibrium. In this context at least, a financialcrisis does not represent a market failure and, hence, does not provide a reasonfor government intervention or regulation.

6.1 COMPLETE MARKETS

Since the time of Adam Smith, economists have been fascinated with the prop-erties of a decentralized market system. Over the last two centuries, they haverefined their theoretical understanding of the conditions that must be satisfiedif the market allocation is to be efficient. The conditions are not innocuous:

• markets must be perfectly competitive;• there must be no externalities;• there must be no asymmetric information (moral hazard or adverse

selection);• there must be no transaction costs;• and markets must be complete.

From our perspective, the critical condition is the completeness of markets.Technically, we say that markets are complete if it is possible to trade, at a singlepoint in time, all the commodities that will ever exist. In defining commod-ities, we distinguish goods that are delivered at different dates or in differentplaces as different commodities. We also distinguish contingent commodities,that is, goods whose delivery is contingent on the occurrence of an uncertainevent. In order for markets to be complete, it must be possible to trade all thecommodities, so-defined, at a single date.

It is both a strength and a weakness of the theory that the definition ofcommodities is very broad. On the one hand, the assumption of completemarkets may seem like a “tall order.” Most economists would not claim that


markets are even approximately complete and this is a weakness of the theory.2

On the other hand, the assumption of complete markets is crucial because itallows us to extend the theory to cover many new phenomena just by defin-ing commodities appropriately. The economic principles that were developedoriginally to explain exchange of ordinary goods in spotmarkets can be used tostudy allocation decisions across space and time and in the face of uncertainty.Conversely, we can characterize many market failures as examples of “missingmarkets.” Finally, the efficiency of economies with complete markets suggeststhat the invention of new markets may provide a remedy for certain marketfailures.

To get an idea of the role of the complete markets assumption in the analysisof efficient markets and lay the groundwork for our later analysis of financialcrises, it is helpful to consider a simple example and ask what a complete setof markets would look like in this case.

Commodities

As usual, we assume there are three dates, t = 0, 1, 2 and a single, all-purposegood at each date. There are two states of nature, denoted by s = H , L. At date0 the state is unknown, although individuals know the true probability πs ofeach state s. The true state is revealed at the beginning of date 1. Since the stateis unknown at date 0, we cannot make the delivery of the good contingent onthe state at date 0; so there is a single commodity at date 0, the good deliveredindependent of the state at date 0. At dates t = 1, 2, the state is known withcertainty so we can define contingent commodities corresponding to eachstate, the good delivered at date t = 1, 2 in state s = H , L. Thus, there are fivecommodities in all, the single noncontingent commodity at t = 0 and the fourcontingent commodities (t , s) = (1,H ), (2,H ), (1, L), (2, L) at the subsequentdates. This commodity space is illustrated in Figure 6.1.

Consumption

Suppose that a consumer has an endowment of one unit of the good at date 0and nothing at dates 1 and 2. In terms of our definition of commodities, the

2 Markets can be effectively complete even if there does not exist a complete set of markets forall possible contingent commodities. Dynamic trading strategies using a limited set of securitiesallow investors to synthesize a much larger set of derivatives. It is possible that in some circum-stances such strategies can effectively complete the set of markets.Whether a good approximationto complete markets is achieved is ultimately an empirical issue which we are unable to resolvehere; but we believe that incompleteness is an important issue even for financial institutions andsophisticated investors.


Commodity (1,H ) Commodity (2,H )

Commodity (1,L) Commodity (2,L)

Commodity 0

t = 0 t = 1 t = 2

s = H

s = L

Figure 6.1. Commodity space with two states and three dates.

endowment consists of a vector e = (1, 0, 0, 0, 0), indicating that the consumerhas one unit of the single commodity at date 0 and none of the contingentcommodities corresponding to dates 1 and 2 and the states H and L. Wesuppose as usual that the consumer only values consumption at dates 1 and 2and that his preferences are represented by a VNM utility function

U (c1) + βU (c2),

where c1 denotes consumption at date 1 and c2 denotes consumption at date2. If we denote consumption at date t in state s by cts then the consumer’sconsumption bundle is described by a vector c = (0, c1H , c2H , c1L , c2L) and theexpected utility of this bundle is

πH {U (c1H ) + βU (c2H )} + πL {U (c1L) + βU (c2L)} ,where πs denotes the probability of state s occurring.

Since markets are complete, the consumer can purchase any consumptionbundle c = (0, c1H , c2H , c1L , c2L) that satisfies his budget constraint at date0. Since only relative prices matter, there is no essential loss of generality inchoosing the good at date 0 as the numeraire and setting its price p0 equalto one. Letting pts denote the price of one unit of the contingent commodity(t , s), we can write the consumer’s budget constraint as

p1Hc1H + p2Hc2H + p1Lc1L + p2Lc2L ≤ p01 = 1.

The right hand side is the value of the consumer’s endowment and the left handside is the value of his consumption bundle. This looks just like the standardbudget constraint and that is one of the great strengths of the Arrow–Debreumodel: by reinterpreting goods at different dates and in different states ofnature as different commodities, we can extend the standard budget constraintand indeed the theory of competitive equilibrium to deal with uncertaintyand time.


Production

In the same way, we can reinterpret the standard theory of production toexplain the allocation of investment. The production technology for this econ-omy consists of the investment opportunities provided by the two assets, theshort asset and the long asset. As usual, one unit of the good invested in theshort asset at date t produces one unit at date t + 1, where t = 0, 1 , and oneunit invested in the long asset at date 0 produces Rs > 1 units of the longasset in state s at date 2; however, since these investment technologies producecontingent commodities, some care is required in their interpretation.

Let’s start with the short asset. A single unit invested at date 0 producesone unit of the good at date 1, independently of the state. In terms of ourdefinition of commodities, one unit of the good at date 0 produces one unit ofthe contingent commodity (1,H ) and one unit of the contingent commodity(1, L). We can represent this technology by the production vector

a0 = (−1, 1, 0, 1, 0),

where the entry −1 denotes an input of one unit at date 0 and the other entriesdenote outputs of contingent commodities.

Investment in the short asset at date 1 is similar, except that the true state isalready known, so both the input and the output are contingent on the state.For example, if one unit is invested at date 1 in state H , the input consists ofone unit of (1,H ) and the output consists of one unit of (2,H ). Thus, we havetwo technologies at date 1, one for each state, represented by the productionvectors

a1H = (0,−1, 1, 0, 0)

and

a1L = (0, 0, 0,−1, 1).

The long asset represents a technology that can only be used at date 0, so itis simpler to analyze. One unit invested at date 0 produces Rs > 1 units at date2 in state s, so the output is a bundle of contingent commodities, namely, RHunits of (2,H ) and RL units of (2, L). This technology can be represented bythe vector

a2 = (−1, 0,RH , 0,RL).

Since each of these production technologies operates subject to constantreturns to scale, in a competitive equilibrium the profits derived from each


process will be zero. If profits were positive, a profit-maximizing firm wouldbe tempted to operate at an infinite scale, which is inconsistent with marketclearing. If the profits per unit were negative, the firm would shut down andearn zero profits. So the firm can only operate at a positive scale if the profitper unit is zero. Because equilibrium profits are zero, it does not matter whoundertakes the different production activities. We can assume that a represen-tative firm does this or we can assume that individual consumers do it. Withcomplete markets there are many degrees of freedom. Here we will assumethat a representative firm makes all production decisions.

The representative firm chooses a level of investment in each of the activitiesdescribed above. Let y0 denote investment in the short asset at date 0, let y1sdenote investment in the short asset at date 1 in state s = H , L, and let x denoteinvestment in the long asset at date 0 . Each activity must yield non-positiveprofits, so

−1 + p1H + p1L ≤ 0, (6.1)

with equality if y0 > 0,

−p1s + p2s ≤ 0, (6.2)

with equality if y1s > 0 for s = H , L, and

−1 + p2HRH + p2LRL ≤ 0, (6.3)

with equality if x > 0. These conditions are necessary and sufficient for profitmaximization.

One interesting point to note is that, even though the true state is unknownwhen all the decisions aremade at date 0, the firmdoes not face any uncertainty.This is because the firmbuys and sells contingent commodities at knownpricesat date 0. For example, in the case of investment in the short asset at date 0, thefirm buys one unit of the good at date 0 as an input and sells one unit of eachof the contingent commodities at date 1 . The cost of the input is 1 and therevenue from selling the outputs is p1H + p1L , so the profit is p1H + p1L − 1.This profit is “realized”at date 0,when the commodities are traded, rather thanat date 1 when the outputs are produced.

Equilibrium

The requirements for competitive equilibrium are that, at the given prices,

(a) each consumer chooses a consumption bundle that maximizes hisexpected utility, subject to his budget constraint;


(b) the representative firm chooses investments in the various activities tomaximize profits; and

(c) the choices of the consumers and the firm are consistent with marketclearing (i.e. demand equals supply).

For the purposes of illustration, let us assume there are two types ofconsumers, A and B with preferences described by VNM utility functionsUA(c1) + βUA(c2) and UB(c1) + βUB(c2). Formally, a competitive equilib-rium consists of a price vector p∗ = (1, p∗1H , p∗2H , p∗1L , p∗2L), a consumptionbundle c i∗ = (0, c i∗1H , c i∗2H , c i∗1L , c i∗2L) for consumers i = A,B and a vector ofinvestments I∗ = (y∗0 , y∗1H , y∗1L , x∗) such that

(a) c i∗ maximizes consumer i’s expected utility subject to his budgetconstraint, for each i = A,B;

(b) I∗ satisfies the zero profit conditions; and

(c) all markets clear.

Since all trading occurs at date 0, market clearing is achieved at date 0 as well.The actual delivery of goods takes place later, but the decisions have alreadybeen made at date 0 and they must be consistent. The market for the good atdate 0 will clear if

x∗ + y∗0 = 2;

the two consumers supply their one-unit endowments and the firm demandsinputs x∗ and y∗0 for investment in the long and short assets, respectively. Atdate 1, there are two contingent commodities, one for each state s. For thecorresponding markets to clear at date 0 requires

cA∗1s + cB∗1s + y∗1s = y∗0 ,

for each s = H , L; here the firm supplies y∗0 units of the good (the payoff fromthe investment in the short asset) and demands y∗1s units to invest in the shortasset and the consumer demands c∗1s units to consume. At date 2, there areagain two contingent commodities, one for each state s. For the correspondingmarkets to clear at date 0 requires

cA∗2s + cB∗2s = y∗1s + Rsx∗,

for each s = H , L; the firm supplies y∗1s + Rsx∗ units of the good (the returnon investment in the long asset at date 0 and the short asset at date 1) and theconsumers demand cA∗

2s + cB∗2s units of the good for consumption.


Example 1 To illustrate the model in the simplest possible way, consider aRobinson Crusoe economy in which there is a single type of consumer withendowment e = (1, 0, 0) and preferences given by the familiar Cobb–Douglasversion of the VNM utility function

U (c1) + βU (c2) = ln c1 + ln c2.

Assume that the states are equiprobable

πH = πL = 0.5

and the returns on the long asset are

(RH ,RL) = (3, 0) .

Since there is a single representative consumer, there is a unique Pareto-efficient allocation for this economy, namely, the allocation that maximizes therepresentative consumer’s expected utility. The first fundamental theorem ofwelfare economics ensures that a competitive equilibrium is Pareto efficient.Using this fact, we can solve for the competitive equilibrium in two stages.First, we solve for the unique efficient allocation. Second, we find the pricesthat support this allocation as an equilibrium.

Suppose that a planner chooses an attainable allocation that maximizesthe expected utility of the representative consumer. He invests in x unitsin the long asset and y units in the short asset. A necessary condition formaximizing expected utility is that, in each state s, the consumption bundle(c1s , c2s) maximizes the consumer’s utility ln c1 + ln c2 subject to the feasibilityconditions

c1s ≤ y ,

and

c1s + c2s ≤ y + Rsx ,

for s = H , L.The allocation of consumption between dates 1 and 2 depends on the value

of Rs , the return to the long asset. If Rs is high (Rsx > y), it is optimal toput c1 = y and c2 = Rsx . If Rs is low (Rsx < y), it is optimal to equalizeconsumption between the two dates, so that c1 = c2 = 1

2

(y + Rsx

). The

consumption functions are illustrated in Figure 6.2.


c2(R )

c1(R )

R

c1(R ) = c2(R )

c

0

Figure 6.2. Consumption at dates 1 and 2 as a function of the return on the long asset.

In state L the return to the long asset is zero, so the planner divides thereturn to the short asset evenly between consumption at date 1 and date 2:

c1L = c2L = y2.

In state H , by contrast, the liquidity constraint will be binding at date 1:

c1H = y , c2H = 3x .

Now that we have guessed the optimal consumption allocation as a functionof the first period investments in the long and short assets, we need to deter-mine the optimal portfolio (x , y). The planner should choose (x , y) so that anextra unit invested in either asset would increase expected utility by the sameamount. So the first-order condition for maximizing expected utility is

1

2

{1

c1H+ 1

c1L

}= 1

2

{3

c2H+ 0

c2L

}.

The left hand side is the increase in expected utility from increasing y . The righthand side is the increase in expected utility from increasing x . Substituting ourhypothesized expressions for consumption into the first-order condition gives

1

2

{1

y+ 2

y

}= 1

2

1

x,

which is satisfied only if

(x , y) =(

1

4,3

4

).


Then we can use the consumption functions to deduce the efficient consump-tion allocation,

c = (c1H , c2H , c1L , c2L) =(y , 3x ,

y

2,y

2

)=(

3

4,3

4,3

8,3

8

).

It remains to find prices that will support this allocation as an equilibrium.The consumer’s equilibrium decision problem is to maximize his expectedutility

1

2{ln c1H + ln c2H } + 1

2{ln c1L + ln c2L}

subject to the budget constraint

p1Hc1H + p2Hc2H + p1Lc1L + p2Lc2L = 1.

From the properties of the Cobb–Douglas utility function, we know that theconsumerwill spend the same share of hiswealth on eachof the four contingentcommodities. Thus,

p1Hc1H = p2Hc2H = p1Lc1L = p2Lc2L = 1

4.

Using the efficient consumption bundle we can solve for the price vector thatwill support the consumer’s choice of this consumption bundle

p = (p1H , p2H , p1L , p2L) =(

1

3,1

3,2

3,2

3

).

In the notation introduced earlier,

I = (y0, y1H , y1L , x) =(

3

4, 0,

3

8,1

4

).

As we have already shown, the firm invests y0 = 34 in the short asset and x = 1

4in the long asset at date 0. At date 1 there is no investment in the short assetin state H because all the returns to the short asset are consumed. At date 1 instate L, by contrast, half of the return to the short asset is stored until date 2because the return to the long asset is zero.

The zero-profit conditions (6.1)–(6.3) corresponding to y0, x , and y1L holdas equations because investment is positive. The zero-profit condition (6.2)corresponding to y1H is an inequality because the investment is zero. In fact,


we can show that all the zero-profit conditions are satisfied exactly. The zero-profit condition for the short asset at date 0, i.e. condition (6.1), is satisfiedbecause

p1H + p1L = 1

3+ 2

3= 1.

The zero-profit condition for the long asset at date 0, i.e. condition (6.3), issatisfied because

RHp2H + RLp2L = 3 × 1

3+ 0 × 1

4= 1.

The zero-profit constraint for the short asset at date 1, i.e. condition (6.2), issatisfied in each state because

p1H = p2H = 1

3and p1L = p2L = 2

3.

Thus, all the conditions for competitive equilibrium are satisfied.

6.2 INTERMEDIATION AND MARKETS

To study the role of financial markets in allowing intermediaries to hedge risks,we use a variant of the model used in the preceding chapters. There are threedates t = 0, 1, 2 and at each date an all-purpose good that can be used forconsumption or investment. There are two assets, the short asset representedby a storage technology that yields one unit of the good at date t + 1 for everyunit invested at date t , and a long asset represented by a constant returns toscale technology that yields R > 1 units of the good at date 2 for every unitinvested at date 0.

As usual, there is a continuum of economic agents at date 0, each of whomhas an endowment of one unit of the good at date 0 and nothing at futuredates. At date 1, each agent learns whether he is an early consumer, who onlyvalues the good at date 1, or a late consumer, who only values the good atdate 2. The probability of being an early consumer is 0 < λ < 1. Ex post, thefraction of early consumers in the economy is assumed to be λ as well.

The main difference between the current model and those used in the pre-ceding chapters lies in the specification of uncertainty. We assume that theeconomy is divided into two regions, labeled A and B. Ex ante the two regionsare identical, with the same number of identical agents and the same assets.

6.2 Intermediation and Markets 165

There are two aggregate states of nature, denoted by HL and LH . Each state isequally likely, that is, each occurs with probability 0.5. In stateHL the fractionof early consumers in region A is λH and the fraction of early consumers inregion B is λL , where 0 < λL < λH < 1. In state LH the fractions are reversed.We assume that

λ = 1

2(λH + λL) ,

so that the fraction of early consumers in each state is λ. This also implies thatthe probability of any investor becoming an early consumer is also λ.

The investors’ attitudes toward risk are represented by a common VNMutility function. If an investor consumes c units of the good at the appropriatedate, his utility is U (c), where U (·) satisfies all the usual properties.

All uncertainty is resolved, as usual, at the beginning of date 1 when the truestate HL or LH is revealed and each investor learns his type, early or late.

The uncertainty and information structure are illustrated in Figure 6.3.

t = 0 t = 1 t = 2

λA = λH > λB = λL

λA = λL < λB = λH

s = (H,L)

s = (L,H)

Figure 6.3. Liquidity shocks of groups A and B in states (H , L) and (L,H ).

6.2.1 Efficient risk sharing

Suppose a central planner was responsible for making the decisions in thiseconomy. Would the division of investors into two regions matter? Evidentlynot. The investors are ex ante identical and the number of early consumers isthe same in each state, so there is no reason why the planner should pay anyattention to the region to which an investor belongs. The planner will allocatea fraction y of the endowment to the short asset and a fraction 1−y to the longasset and offer c1 units of the good at date 1 to the early consumers and c2 unitsof the good at date 2 to the late consumers, independently of the state. Thesevariables are chosen to maximize the expected utility of the representative


investor

λU (c1) + (1 − λ)U (c2),

subject to the usual feasibility conditions

λc1 = y

and

(1 − λ)c2 = (1 − y)R.

As usual, the first best is characterized by the feasibility conditions and by thefirst-order conditionU ′(c1) = RU ′(c2). Since the first-order condition impliesthat c1 < c2, the incentive constraint is also satisfied, in case the planner doesnot directly observe the investor’s type. The important point is that the optimalconsumption allocation for each individual does not depend on the state. Itonly depends on whether the individual is an early or a late consumer.

Now suppose that an intermediary wanted to implement the first-best out-come. There would be no difficulty if the intermediary served a representativesample of investors from Region A and Region B, for in that case the fractionof early consumers would be exactly λ in each state and the first-best allocationwould clearly be feasible for the intermediary. A more interesting case is onein which intermediaries are heterogeneous, that is, have different proportionsof investors drawn from Region A and Region B. In that case, the proportionof early consumers for a given intermediary may not be certain and equal to λ

and, as a result, the first best may not be attainable.To illustrate the problem, consider the extreme case where an intermediary

is allowed to operate in either region, but not both. Legal restrictions of thisform were common in the United States in the past. Even in the present day,there are still many US banks that, because of their size, operate only within asmall region. As usual, we assume that there is free entry and that competitionforces intermediaries to maximize their depositors’ expected utility.

The problem for the intermediary is that the fraction of early consumersamong his depositors will now vary with the aggregate state. If the fraction ofearly consumers is high, then

λHc1 > λc1 = y .

There is not enough liquidity at date 1 to provide the early consumers with thepromised amount of consumption. Similarly, if the fraction of early consumers


is low, then

(1 − λL)c2 > (1 − λ)c2 + (λ − λL)c1

= (1 − y)R + (y − λLc1).

The intermediary has toomuch liquidity at date 1 and not enough to distributeat date 2. So, clearly, the intermediary cannot achieve the first best in autarky.

6.2.2 Equilibrium with complete financial markets

The intermediary’s problem is easily solved if we introduce markets in whichthe intermediary can insure against getting a high liquidity shock. Since thereis no aggregate uncertainty, whenever some intermediaries get a high liquidityshock and hence have too little liquidity to give their depositors the first-bestconsumption c1, there will be intermediaries in the other region with a lowliquidity shock and too much liquidity at date 1. These intermediaries wouldbe happy to lend to the liquidity-constrained intermediaries in order to getmore consumption at date 2 when they will need it.

In order to implement the first best, we need complete markets. There arefive contingent commodities, the good at date 0, the good at date 1 in statesHL and LH , and the good at date 2 in states HL and LH . Taking the goodat date 0 as the numeraire (p0 ≡ 1), let pts denote the price of the good atdate t in state s, where t = 1, 2 and s = HL, LH . The no-arbitrage conditionfor the short asset at date 0 implies that p1HL + p1LH = 1. The symmetryof the two states suggests that there exists a symmetric equilibrium in whichp1HL = p1LH . Putting together these two conditions gives

p1HL = p1LH = 1

2.

Similarly, we assume that p2HL = p2LH and use the no-arbitrage condition forthe long asset,

(p2HL + p2LH

)R = 1, to conclude that

p2HL = p2LH = 1

2R.

Then without risk of confusion we can write p1 for the price of the goodat date 1 and p2 for the price of the good at date 2 in either state. Supposethe intermediary in Region A promises a consumption profile (c1, c2) that is


independent of the state. The intermediary’s budget constraint is

p1HLλHc1 + p1LHλLc1 + p2HL (1 − λH ) c2 + p2LH (1 − λL) c2 = 1

⇐⇒ p1 (λH + λL) c1 + p2 (1 − λH + 1 − λL) c2 = 1

⇐⇒ λc1 + 1

R(1 − λ) c2 = 1.

The analogous calculation for intermediaries in Region B yields an identicalbudget constraint. Then each intermediary is choosing a profile (c1, c2) tomaximize the expected utility λU (c1) + (1 − λ)U (c2) subject to the budgetconstraint above and this requires that the first-order condition U ′ (c1) =RU ′ (c2) be satisfied. In other words, the first-best consumption profile satisfiesthe intermediary’s budget constraint and satisfies the first-order condition,hence is optimal. It is also easy to check that the zero-profit conditions aresatisfied.

This allocation is feasible for the economy, as we have already seen in ouranalysis of the planner’s problem. Somarketswill clearwhen each intermediarychooses the first-best consumption profile and the corresponding productionplan. Thus, complete markets allow precisely the transfers between states thatare necessary for first-best risk sharing.

Example 2 In the preceding sketch of the intermediary’s problemwe assumedthat consumers received the same consumption independently of the state.Wewill obtain this condition as part of the solution to the intermediary’s problemusing the following parameters:

R = 3;

U (c) = −1

5c−5;

λH = 0.6, λL = 0.4;

πHL = πLH = 0.5.

Suppose that

p1HL = p1LH = p1 = 1

2

and

p2HL = p2LH = p2 = 1

6.


We look at the problem from the perspective of an intermediary in Region Aand let (c1H , c2H ) (resp. (c1L , c2L)) denote the consumption profile promisedwhen the fraction of early consumers is λH (resp. λL) for that intermediary.The intermediary needs to maximize the representative depositor’s expectedutility

1

2{0.6U (c1H ) + 0.4U (c2H ) + 0.4U (c1L) + 0.6U (c2L)}

subject to the intermediary’s budget constraint

p1 (0.6c1H + 0.4c1L) + p2 (0.4c2H + 0.6c2L) = 1.

The first-order conditions for this problem are

U ′ (c1H ) = U ′ (c1L) = µp1 = µ

2

U ′ (c2H ) = U ′ (c2L) = µp2 = µ

6

where µ is the Lagrange multiplier for the budget constraint. Substituting theformula for marginal utility into the first-order conditions we get

(c1H )−6 = (c1L)−6 = µp1 = µ

2

(c2H )−6 = (c2L)−6 = µp2 = µ

6.

Then c1H = c1L = c1 and c2H = c2L = c2, that is, consumption is independentof the state, as we should expect in an efficient allocation, and we can solvethese first-order conditions for the optimal consumption ratio

c2c1

= 6√

3 = 1.201.

Using the relationship between c1 and c2 and the budget constraint we cansolve for

p1 (0.6c1 + 0.4c1) + p2 (0.4c2 + 0.6c2) = p1c1 + p2c2= 1

2c1 + 1

6c2

= 1

2c1 + 1

6(1.201) c1 = 1.


This equation can be solved for c1

c1 = 1.428

and then we use the optimal ratio to find c2

c2 = 1.201c1 = (1.201) (1.428) = 1.715.

6.2.3 An alternative formulation of complete markets

An alternative to assuming a complete set of markets for contingent commod-ities at date 0 is to allow trade to occur sequentially. As described in Section6.1, markets are sequentially complete, in the simple, two-state, three-periodmodel, if there are twoArrow securities at date 0 and spot and forwardmarketsfor the good at date 1.We first describe the spot and forward markets at date 1,after all uncertainty has been resolved, and then we describe theArrow securitymarkets at date 0.

At date 1 we assume there is a forward market in which intermediaries cantrade the good at date 1 for promises to deliver the good at date 2. Let p denotethe value of one unit of the good at date 2 in terms of units of the good at date1. In other words, p is the present value of one unit of the good at date 2. Sinceone unit of the long asset yields R units of the good at date 2, the value of oneunit of the asset at date 1 will be pR.

Suppose that an intermediary has a portfolio, including trades in Arrowsecurities made at date 0, that is worth ws in the state where the liquidity shockis λs , s = H , L. The intermediary’s budget constraint is

λs c1 + p(1 − λs)c2 = wsand the intermediary will choose (c1, c2) to maximize

λU (c1) + (1 − λs)U (c2)

subject to this budget constraint. Let V (p,ws ; λs) denote the maximized valueof this utility function.

In Section 6.1 we assumed that the forward price p was state-dependent.Here the two states are different from the point of view of an individual inter-mediary but identical from a macroeconomic point of view. In other words,the proportions of early consumers in the intermediary vary from state tostate but the proportion of early consumers in the economy as a whole is thesame. Since the two states that can occur at date 1 are identical from an aggre-gate point of view, we consider a symmetric equilibrium in which the price


p is independent of the state. This means that the returns to the two assetsare certain at date 0. In order to persuade intermediaries to hold both assets,the one-period returns must be equal, which, as we have seen in Chapter 4,requires p = 1/R. Then the value of the intermediary’s portfolio at date 1 isindependent of the amount invested in the short and long assets.

Now consider the intermediary’s problem at date 0. An Arrow security atdate 0 promises delivery of one unit of the good in one state at date 1 andnothing in the other state. Since we are considering a symmetric equilibrium,the prices of the two securities should be equal, and without loss of generalitywe can normalize them to equal 1. Let zs be the amount of the Arrow securitythat pays off in the state where the intermediary’s liquidity shock is λs . Notethat s refers to the intermediary’s state, not the aggregate state, but since thetwo are perfectly correlated, this should not lead to any confusion. Withoutloss of generality, we can assume that the intermediary invests all of its depositsin the short and long assets, so the trades in Arrow securities must balance,that is,

zH + zL = 0.

In other words, the intermediary buys Arrow securities in one state and sellsthem in another. The value of the intermediary’s portfolio at date 1 will be

ws = y + p(1 − y)R + zs= 1 + zs

for s = H , L. Clearly, the intermediary can attain any pattern of date-1 payoffs(wH ,wL) that satisfy wH + wL = 2. So the intermediary should choose aportfolio of Arrow securities (zH , zL) to maximize the expected value of theindirect utility function ∑

s

V (p,ws ; λs)


1

2(wH + wL) = 1.

The solution of this problem implies that the marginal utility of consumptionmust be the same in each state, that is, wH = wL , which in turn implies thatthe consumption allocation (c1, c2) will be the same in each state.

The importance of this alternative approach using sequential trade is thatit makes clear (a) that complete markets for contingent commodities are not


strictly necessary formarkets to be effectively complete and (b) one can achievethe same results with fewer markets. In general, if there are S states and T + 1periods and uncertainty is resolved at date 1, there are ST + 1 contingentcommodities and hence ST + 1 markets in the Arrow–Debreu model. Thesequential trade model, by contrast, requires S Arrow securities and marketfor the T + 1 dated commodities in each state, so the number of marketsis S + T + 1. When the number of states and dates is large (or there aremany physical goods in each date and state), the difference between the twoapproaches becomes even greater.

6.2.4 The general case

The two-state model developed in Section 6.2 is quite special, not only becauseit is restricted to two states, HL and LH , but especially as regards the lack ofaggregate uncertainty. The argument based on this special case is quite general,however. Allen and Gale (2004) present a general model of an economy withfinancial markets for aggregate risks and intermediaries and establish similarresults. It is beyond the scope of this chapter to do more than give some of theflavor of the Allen–Gale analysis. We can do this by extending the two-statemodel to a more general environment with aggregate uncertainty about bothasset shocks and liquidity shocks. To simplify the notation, we continue toassume that there are two regions, A and B, that the probability of being anearly consumer may be different in each region, but that the asset returns arethe same in each region.

There is assumed to be a finite number of states indexed by s = 1, . . . , S andeach state s occurs with probability πs > 0. There are three dates, t = 0, 1, 2,the state is unknown at date 0 and the true state is revealed at the beginningof date 1. At date 1, each agent learns whether he is an early or late consumer.The utility of consumption is represented by a commonVNM utility functionU (c). The probability of being an early consumer can depend on both thestate s and the region i. We let λi(s) denote both the probability of being anearly consumer and the proportion of early consumers in region i in state s.There are two assets, short and long. The return to the short asset is alwaysone, independently of the state, but the return to the long asset may depend onthe state. We let R(s) denote the return to one unit invested in the long asset ifthe state is s.

Markets play an essential role only if there is heterogeneity among intermedi-aries; otherwise there are no gains from trade. We focus on the extreme casewhere there are distinct intermediaries for regions A and B. An intermediaryof type i draws all its depositors from region i. Since the proportion of early


consumers can be different in each region, intermediaries can insure the risk ofliquidity shortages by trading on markets for contingent commodities, buyingliquidity in states where it expects high demand and selling it in states where itexpects low demand for liquidity. The sharing of risk is intermediated by themarkets, so the risk sharing across regions is not immediately apparent. Whatthe intermediary appears to be doing is demanding an optimal consumptionprofile for its depositors, subject to a budget constraint; however, the effect isthe same as if the intermediaries in different regions were writing optimal risk-sharing contracts with each other, promising to share the available liquidity inthe way a central planner would.

The intermediary takes an endowment of one unit from each agent at date 0and offers in exchange c1(s) units of the good in state s to everyone who with-draws at date 1 and c2(s) units of the good in state s to everyone whowithdrawsat date 2. Since the intermediary cannot tell whether the person withdraw-ing is an early or a late consumer, the intermediary has to ensure that lateconsumers have no incentive to pretend that they are early consumers andwithdraw at date 1. So we assume that every consumption plan satisfies theincentive constraint

c1(s) ≤ c2(s) for s = 1, . . . , S. (6.4)

A consumption plan c = {(c1(s), c2(s))}Ss=1 specifies a consumption profilec(s) = (c1 (s) , c2 (s)) for every state s. The consumption plan is calledincentive-compatible if it satisfies the incentive constraint (6.4).

There is a single commodity at date 0 and a contingent commodity forevery state s = 1, . . . , S and date t = 1, 2. Let the commodity at date 0 bethe numeraire and let p1(s) denote the price of the good in state s at date 1and p2(s) denote the price of the good in state s at date 2. The intermediaryis assumed to be able to trade all the contingent commodities at date 0. Theintermediary receives one unit of the commodity at date 0 and uses this topurchase the contingent commodities that it has promised to the depositors.The intermediary’s budget constraint can be written as

S∑s=1

{p1(s)λi c1(s) + p2(s) (1 − λi (s)) c2(s)

} ≤ 1, (6.5)

where the left hand side is the cost of the consumption plan and the right handside is the value of a single agent’s deposit.

Free entry and competition require the intermediary to maximize theexpected utility of the representative depositor, so the intermediary’s decisionproblem is to choose an incentive compatible consumption plan c = {c (s)} to


maximize the expected utility of the representative depositor

S∑s=1

πs {λi(s)U (c1(s) + (1 − λi(s)U (c2(s))}

subject to the budget constraint (6.5).As before, the existence of a complete set of markets makes the physical

assets redundant in the sense that an intermediary does not need to hold them.Someone must hold them, however, to produce outputs of the goods at dates 1and 2 but we can assume that a representative firm chooses to invest y0 in theshort asset at date 0, x in the long asset at date 0 and y1 (s) in the short assetat date 1 in state s. Since these investments earn zero profits in equilibrium,it really does not matter who makes the investments. Let I = (y0, {y1(s)} , x)denote the vector of investments undertaken.

The zero-profit conditions are analogous to the oneswederived before. Sinceinvestment in the assets is subject to constant returns to scale, positive profitsare inconsistent with equilibrium and positive investment will occur only ifthe profits are non-negative (i.e. zero). One unit invested in the short asset atdate 0 yields one unit at date s in each state, so the zero-profit condition is

S∑s=1

p1(s) ≤ 1, (6.6)

with equality if y0 > 0. The left hand side of (6.6) is the value of the outputs atdate 1 and the right hand side is the value of the input at date 0. The inequalitystates that investing in the short asset at date 0 yields non-positive profits andthe profits must be zero if investment is positive.

Similarly, one unit invested in the short asset at date 1 in state s yields oneunit at date 2 in state s, so the zero-profit condition is

p2(s) ≤ p1(s), (6.7)

with equality if y1(s) > 0, for s = 1, . . . , S. Again, the left hand side of (6.7) isthe value of outputs and the right hand side is the value of inputs. Finally, oneunit invested in the long asset at date 0 yields R(s) units at date 2 in state s, sothe zero-profit condition is

S∑s=1

p2(s)R(s) ≤ 1, (6.8)

with equality if x > 0.


All contingent commodities are traded at date 0. Subsequently, intermedi-aries simply fulfill the commitments they entered into at date 0. If the marketsclear at date 0, the subsequent execution of these trades must be feasible. First,consider the market for the commodity delivered at date 0. The supply ofthe commodity is equal to the endowment of the good. The demand for thecommodity equals the investment in the short and long asset at date 0. Thenmarket clearing requires

x + y0 = 1. (6.9)

There is a contingent commodity for each state s at date 1. The demand forthis commodity equals the total consumption of the early consumers in the tworegions plus the investment in the short asset. The supply of this commodityequals the output from the investment in the short asset at date 0. Thenmarketclearing requires

1

2{λA (s) cA1(s) + λB (s) cB1(s)} + y1(s) = y0, (mc2)

for each s = 1, . . . , S. Note that on the left hand side the total demand forconsumption is the average of consumption in the two regions, because halfthe population is in each region.

There is a contingent commodity for each state s at date 2. The demandfor this commodity is equal to the total consumption of the late consumersfrom the two regions (there is no investment at date 2). The supply is theoutput from the investment in the short asset at date 1 plus the output fromthe investment in the long asset at date 0. Market clearing requires

1

2{(1 − λA (s)) cA2(s) + (1 − λB (s)) cB2(s)} = R(s)x + y1(s), (6.10)

for each s = 1, . . . , S.In the economy just described, the incentive constraint (6.4)may be binding.

If it is binding, then the first-best allocation (the allocation a central plannerwould choose if he had complete information about the agents’ types) maynot be feasible.We take the view that a regulator or planner has no more infor-mation than the market and so the consumption plans implemented by theplanner must also satisfy the incentive constraint. In that case, the first best isnot the appropriate benchmark. We should instead ask: “What could a centralplanner achieve if he had the same information as the market and were subjectto the same incentive constraint?” This sort of reasoning leads to the con-cept of incentive efficiency. An allocation consisting of the consumption plans


c = (cA , cB) and the investment plan I is attainable if it satisfies the market-clearing conditions (6.9)–(6.10). An attainable allocation (c , I ) is incentivecompatible if the consumption plans satisfy the incentive constraint (6.4). Anincentive-compatible allocation (c , I ) is incentive efficient if there does notexist an incentive-compatible allocation

(c ′, I ′)

that makes both regions ofinvestors better off ex ante. In other words, the planner cannot do better thanthe market when the planner has the same information as the market.

Allen and Gale (2004) show that any equilibrium allocation is incentiveefficient as long as markets are complete and investors are restricted to usingintermediaries to access the markets. We sketch the argument here. Supposethat (c∗, I∗) is an equilibrium allocation and p∗ is the equilibrium price vector.If both regions can be made better off by choosing an incentive-compatibleallocation (c , I ), then it must be the case that cA and cB are not within theintermediaries’ budget sets, that is,

S∑s=1

{p∗1 (s)λi c∗i1(s) + p∗2 (s) (1 − λi (s)) c

∗i2(s)}

> 1 (6.11)

for each i = A,B. For all the investment activities, either profits are zero or theinvestment level is zero. This implies that the sum of profits is zero:(

S∑s=1

p∗1 (s) − 1

)y0 +

S∑s=1

(p∗2 (s) − p∗1 (s)

)y1(s)

+(S∑s=1

p∗2 (s)R(s) − 1

)x = 0. (6.12)

Rearranging this equation we get

S∑s=1

p∗1 (s)(y0 − y1(s)

)+ S∑s=1

p∗2 (s)(R(s)x + y1 (s)

) = x + y0, (6.13)

and substituting from the market-clearing conditions (6.9)–(6.10) intoequation (6.13) we have

S∑s=1

p∗1 (s)(

1

2{λA (s) cA1(s) + λB (s) cB1(s)}

)

+S∑s=1

p∗2 (s)(

1

2{(1 − λA (s)) cA2(s) + (1 − λB (s)) cB2(s)}

)= 1


or

1

2

∑i=A,B

S∑s=1

{p∗1 (s)λi (s) ci1(s) + p∗2 (s) (1 − λi (s)) ci2(s)

} = 1

contradicting the inequality (6.11).

6.2.5 Implementing the first best without complete markets

In very special cases, it may be possible to implement the first best withoutcomplete markets. To see this, we return to the special model with two regions,A and B, and two states,HL and LH , and suppose that there is an asset marketat date 1. An intermediary could sell some of the long asset in order to getadditional liquidity in the “high” state and buy the long asset with the excessliquidity in the “low” state. In the special case of logarithmic utility and noaggregate uncertainty, this is enough to achieve the first best. Suppose theVNM utility function is

U (c) = ln(c)

and the planner’s problem is to maximize

λ ln(c1) + (1 − λ) ln (c2)

subject to the constraints

λc1 = y(1 − λ) c2 = R (1 − y) .

The solution to this problem is the allocation that gives early consumersc∗1 = 1 and late consumers c∗2 = R. To achieve this consumption profile,the investment in the short asset at date 0 must be y∗ = λ. Then

λc∗1 = λ = y∗

and

(1 − λ) c∗2 = (1 − λ)R = R (1 − y∗) .This is the allocation that a central planner, who could move goods betweenthe regions, would implement. Can the intermediaries do the same by tradingon the asset market at date 1?


In the absence of aggregate uncertainty (remember, intermediaries fromboth regions trade in the economy-wide asset market) the asset price at date 1must be P = 1. Otherwise, intermediaries would not be willing to hold bothassets at date 0. Suppose that the intermediaries invest y = λ in the short assetand 1 − y in the long asset and promise the early consumers c1 = 1 at date 1.Then at date 1, an intermediary in the high state will need λHc1 = λH units ofthe good but only has y = λ units of the short asset. So it needs an additionalλH − λ units of the good. To get this, it must sell λH − λ units of the longasset (remember the asset price is P = 1). An intermediary in the low state hasy = λ units of the short asset but only needs λLc1 = λL units of the good topay the early consumers. Since P < R, the short asset is dominated at date 1and no one wants to hold it between date 1 and date 2. All of the intermediary’sexcess liquidity λ − λL will be supplied in exchange for the long asset. So thesupply of the long asset is λH − λ, the demand is λ − λL , these two amountsare equal because λ = 1

2 (λH + λL), and the market clears at date 1.What happens at the last date? In the high state, the intermediary has

(1 − y) − (λH − λ) = (1 − λ) − (λH − λ) = 1 − λH

units of the long asset and so can provide R units to each of the 1 − λH lateconsumers. Similarly, an intermediary in the low state has

(1 − y) + (λ − λL) = (1 − λ) + (λ − λL) = 1 − λL

units of the long asset, and so can provide R units of the good to each ofthe 1 − λL late consumers. So the intermediaries can achieve the first-bestallocation of risk sharing in each state and this is clearly the best that theycan do.

For the economy as a whole there is no change in the proportion of earlyconsumers and so efficient risk sharing requires that an individual’s consump-tion should depend on whether he is an early or late consumer, but not on theproportion of early consumers in his region. This is only possible because themarket allows liquidity to be reshuffled between regions. Regions with excessliquidity supply it to regions with insufficient liquidity in exchange for thelong asset. A region with a high proportion of early consumers will have a lowproportion of late consumers and so needs less of the long asset to provideconsumption to late consumers. A region with a low proportion of early con-sumers has a high proportion of late consumers and needs more of the longasset. So these trades satisfy everyone’s needs.

This result is special and relies both on the absence of aggregate uncertaintyand the assumption of logarithmic utility. A couple of numerical examples will


serve to show what is special about the case of logarithmic utility and whatgoes wrong when we have incomplete markets with other preferences.

Example 3 To illustrate the implementation of the first best with logarithmicutility, suppose that

R = 3;

U (c) = ln(c);

λH = 0.6, λL = 0.4.

Then the efficient allocation is given by c∗1 = 1, c∗2 = 3, and y∗ = λ = 0.5.At t = 1, an intermediary in the high state provides λHc∗1 = 0.6 units of

consumption to early consumers, but only receives y∗ = 0.5 units from theshort asset. It therefore has to sell 0.1 units of the long asset to make up thedifference. At t = 2, it now has

1 − y∗ − 0.1 = 0.5 − 0.1 = 0.4

units of the long asset. The proportion of late consumers is

1 − λH = 1 − 0.6 = 0.4,

so each consumer gets

0.4R

(1 − λH )= 0.4 × 3

0.4= 3

units of consumption, as required.At t = 1, an intermediary in the low state provides

λLc∗1 = 0.4 × 1 = 0.4

units of consumption to early consumers, but receives y∗ = 0.5 units fromthe short asset. It therefore has 0.1 units of the good to sell in exchange for thelong asset. At t = 2, it now has

1 − y∗ + 0.1 = 0.5 + 0.1 = 0.6

units of the long asset. The fraction of late consumers is 1−λL = 1−0.4 = 0.6,so each consumer gets

0.6R

(1 − λL)= 0.6 × 3

0.6= 3

units of consumption, as required.


Example 4 Now suppose that the parameters of the example are the sameexcept that the utility function is assumed to be

U (c) = −1

5c−5, (6.14)

that is, the constant coefficient of relative risk aversion is six. The optimalconsumption allocation (for a central planner) satisfies U ′(c∗1 ) = RU ′ (c∗2 )and the feasibility condition λc∗1 + (1 − λ) c∗2 /R = 1. These conditions can besolved to give us the optimal consumption allocation

c∗1 = 1.428

and

c∗2 = 1.715.

The central planner can implement this consumption allocation by choosing

y∗ = λc∗1 = (0.5) (1.428) = 0.714.

Note that at t = 1 the long asset will produce

R(1 − y∗) = (3) (0.286) = 0.858

and the late consumers receive a total of

(1 − λ) c∗2 = (0.5) (1.715) = 0.858.

If an intermediary tries to implement the same allocation, at t = 1 in the highstate the intermediary needs to give the early consumers

λHc∗1 = (0.6) (1.424) = 0.854

but only has

y∗ = 0.714

from the short asset. The intermediarywill have to obtain the difference 0.854−0.714 = 0.140 by selling 0.140 units of the long asset. Then at t = 1, theintermediary will have(

1 − y∗)− 0.140 = 0.286 − 0.140 = 0.146

6.3 Incomplete Contracts 181

units of the long asset, which will yield a return of

(3) (0.146) = 0.437 < 0.858 = (1 − λ) c∗2 .

In other words, the intermediary in the high state will not be able to give thelate consumers c∗2 if it gives the early consumers c∗1 . The intermediary cannotimplement the first best.

In Examples 3 and 4, the intermediary wealth at date 1 is always w = 1,regardless of the portfolio chosen, because P = 1.With logarithmic utility, theoptimal consumption profile gives early and late consumers the same presentvalue of consumption. If the proportion of early consumers changes, the bud-get is still balanced because in present value terms the cost of early and lateconsumers is the same. When relative risk aversion is greater than one, theoptimal consumption profile gives early consumers a higher present value ofconsumption (but a lower level of actual consumption) than late consumers.So if the proportion of early consumers goes up, the cost of the consumptionprofile increases. So consumption at both dates must be reduced to balance thebudget. Without markets to allow intermediaries to make trades contingenton the state, there is no way the intermediary can achieve the first best. Thelogarithmic case is the unique case (with constant relative risk aversion) inwhich complete markets are not required.

Another way to think of the problem is to argue that, although there areenough assets and goods in the economy to give everyone the first-best con-sumption allocation, the price of liquidity is too high (the price of the long assetis too low). The intermediary in the high state has to give up too much of thelong asset in order to get the liquidity it needs at t = 1 and so finds itself shortof goods at t = 2. The intermediary in the low state, by contrast, does very wellout of this state of affairs. It gets a large amount of the long asset in exchange forits supply of liquidity at t = 1 and can give its customersmore than c∗2 at t = 2.

Although there are gainers and loses ex post, from the point of view of anintermediary at t = 0, uncertain whether it will be in the high or the low state,the gains in the low state do not compensate for the losses in the high state. Theintermediary’s depositors are unambiguously worse off than in the first best.

6.3 INCOMPLETE CONTRACTS

There is no aggregate uncertainty in the simple two-state economy discussedin Sections 6.2.1 and 6.2.2. As a result, the first-best allocation of consumptiondoes not depend on the state of nature. This is a rather special case, however.


In general, we should expect the first-best allocation will depend on every statein a more or less complicated way. Then the first best can only be achievedby a risk-sharing contract that is contingent on every state of nature. Wecall such contracts complete. In the preceding sections, we have assumed thatintermediaries can make consumption contingent on the state, even if they didnot choose to do so. In this sense, we implicitly assumed that contracts werecomplete. In practice, we often observe intermediaries using much simplercontracts. An example is the familiar deposit contract, where an intermediarypromises its depositors d1 units of consumption if they withdraw at date 1 andd2 units if they withdraw at date 2, independently of the state of nature. Wecall this contract incomplete because the promised amounts are required to beindependent of the state.

Incomplete contracts place an additional constraint on the set of feasibleallocations and may prevent the achievement of the first best. If the firstbest is not attainable, it is tempting to say that there is a market failure,but we should remember that there are reasons why contracts are incom-plete. For example, transaction costs makes it costly to write and implementcomplete contracts. We take the view that these costs impose a constraint onthe planner as well as on market participants. Then the appropriate bench-mark is not the first best, but rather the best that a planner could do subjectto the same costly contracting technology. We should ask: “What is the bestthat can be achieved using the available contracting technology?” We will seethat as long as markets are complete, the use of incomplete contracts doesnot lead to market failure: the laisser-faire equilibrium is constrained effi-cient, in the sense that a planner cannot do better using the same contractingtechnology.

6.3.1 Complete markets and aggregate risk

To illustrate, we adapt the model of Sections 6.2.1 and 6.2.2 by assuming thatthere is a single type of investor ex ante and that, ex post, investors are earlyconsumers with probability λ, independently of the state. We also assume thatthe returns to the long asset are stochastic. More precisely, there are two equi-probable states, H and L, and the return is RH in state H and RL in state L.Because there is a single type of investor ex ante, a single intermediary can actlike a central planner and implement the optimal allocation without resortingto trade in markets. In other words, markets are trivially complete.

If the intermediary can offer a complete contingent contract c =(c1H , c2H , c1L , c2L) then it will achieve the first best. The intermediary choosesa portfolio (x , y) and consumption allocation c = (c1H , c2H , c1L , c2L) to


maximize the expected utility of the typical depositor

1

2{λU (c1H ) + (1 − λ)U (c2H )} + 1

2{λU (c1L) + (1 − λ)U (c2L)}

subject to the feasibility constraints

λc1s ≤ yand

λc1s + (1 − λ) c2s ≤ y + Rsx ,for s = H , L.

Now suppose that the intermediary is constrained to use an incompletecontract. For example, suppose that he offers a deposit contract (d1, d2) tothe investors who deposit their endowments. As usual, we assume that d2 ischosen large enough that it always exhausts the value of the assets at date2. In other words, the late consumers always receive the residual value of theintermediary’s assets. Thenwe can forget about the value of d2 and characterizethe deposit contract by the single parameter d1 = d .

In the absence of default, the intermediary must choose a portfolio (x , y)and consumption bundle c = (c1H , c2H , c1L , c2L) such that c1H = c1L = d . Theefficient incomplete contract must solve the problem of maximizing expectedutility

1

2{λU (d) + (1 − λ)U (c2H )} + 1

2{λU (d) + (1 − λ)U (c2L)}

subject to the feasibility constraints

λd ≤ yand

λd + (1 − λ) c2s ≤ y + Rsx ,for s = H , L.

The incompleteness of contracts imposes a welfare cost on investors, sincetheir consumption is constrained to be the same in each state, independentlyof the costs and benefits of consumption. This raises the possibility that defaultmight be optimal, since default allows for a greater degree of contingency inconsumption. For example, suppose that the optimal consumption is higher


in stateH than in state L. In the absence of default, a deposit contract implies aconsumption allocation that is a compromise between the first-best consump-tion levels, being too low in state H and too high in state L. It might be betterfor investors to offer the first-best consumption in state H , that is, to choose(d1, d2) = (c1H , c2H ) and default in state L. If the intermediary defaults on thecontract in state L, the portfolio is liquidated and the early and late investorsshare the liquidated value of the portfolio. Default allows the level of con-sumption to vary between the states but there is another constraint: in thedefault state, the level of consumption has to be equal at each date. Thus, theintermediary will choose (x , y) and c = (c1H , c2H , c1L , c2L) to maximize

1

2U (y + RLx) + 1

2{λU (d) + (1 − λ)U (c2H )}

subject to the constraint

λd + (1 − λ) c2H = y + RHx .

Which of these provides the higher level of expected utility depends on allthe parameters and prices. The crucial point for the current discussion is that,with complete markets, the intermediary makes the correct choice. Clearly, wecannot hope for the first best to be achieved, because the consumption planis constrained by the incompleteness of the contract and by the bankruptcycode. However, subject to these constraints, a planner can do no better thanthe intermediary. More precisely, an equilibrium is defined to be constrainedefficient if there does not exist a feasible allocation that can be implementedusing a deposit contract and that makes some agents better off and no oneworse off. The reason is essentially the same as in the case studied in Section 6.2:if there is a preferred choice for the intermediary it must be outside the budgetset and if every intermediary violates its budget constraint the attainabilitycondition cannot be satisfied. Thus, whether or not the intermediary choosesto default in equilibrium, there is no market failure. The incidence of financialcrises is constrained efficient.

Allen and Gale (2004) have proved the constrained-efficiency of laisser-faireequilibrium under general conditions. The importance of this result is thatit shows that financial crises do not necessarily constitute a market failure.There are welfare losses associated with the use of incomplete contracts, to besure, but these losses are present whether there is a crisis or not. The crucialobservation is that given the contracting technology, the market achieves thebest possible welfare outcome. A planner subject to the same constraints couldnot do better.


Example 5 We have argued that, as long as markets are complete, anintermediary will choose the optimal incidence of default. To illustrate thepossibilities we use a numerical example. Suppose that each investor has anendowment e = (1, 0, 0) and preferences are given by the VNM utility

U (c1) + βU (c2) = ln c1 + ln c2.

Assume that the states are equiprobable

πH = πL = 0.5,

the proportion of early consumers is

λ = 0.5,

and the returns on the long asset are

(RH ,RL) = (3, 0) .

With no default, the consumption allocation is

c = (c1H , c2H , c1L , c2L) = (d , c2H , d , d),

where d = y and c2H = 2(RHx + y/2) = 6x + y . The first-order condition

that determines the optimal portfolio is

1

2

{1

y+ 1

y

}= 1

2

{3

6x + y + 0

y

},

which implies that x = 112 and y = 11

12 . Then the consumption allocation is

c = (y , 6x + y , y , y) =(

11

12,17

12,11

12,11

12

)

and expected utility is

1

4{U (c1H ) + U (c2H ) + U (c1L) + U (c2L)} = 1

4

{3 ln

11

12+ ln

17

12

}

= 1

4× 0.087.

Now suppose that we allow for default. Nothing changes in the low state,because the only asset that has value is the short asset and this is already


being divided evenly between the early and late consumers. In the high state,by contrast, the first-best allocation can be achieved. This requires giving theearly consumers 2y and the late consumers 2RHx = 6x . Then the first-ordercondition for the optimal portfolio is

1

2

{1

2y+ 1

y

}= 1

2

{3

6x+ 0

y

},

which implies that x = 14 and y = 3

4 . Then

c = (2y , 6x , y , y) =(

3

2,3

2,3

4,3

4

)

and the expected utility is

1

4{U (c1H ) + U (c2H ) + U (c1L) + U (c2L)}

= 1

4

{2 ln

3

2+ 2 ln

3

4

}= 1

4× 0.236.

Clearly, default increases expected utility here because default relaxes thedistortion of the deposit contract (the requirement that consumption at date1 be the same in each state) without introducing an offsetting distortion (inthe default state, consumption is equal at both dates, whether there is defaultor not) as in Zame (1993).

6.3.2 The intermediary’s problem with incomplete markets

Example 3 showed us that the case of log utility is very special. It has theproperty that, at the first best, the present value of consumption is the samefor early and late consumers, that is,

c∗1 = 1

Rc∗2 .

This is not true for most utility functions. If c∗1 > c∗2 /R, as in the case of theutility function (6.14) in Example 4, then the present value of consumptionfor the early consumers is greater than the present value of consumptionfor the late consumers. An increase in the proportion of early consumers willincrease the present value of total consumption, so in order to satisfy the budgetconstraint the intermediary will have to reduce someone’s consumption. Sinceearly consumers are promised a fixed amount d , the intermediary will end up


giving less to the late consumers, assuming it can do so without causing a run.Assuming that c1 = d , the late consumers receive c2s in state s = H , L, where

c2s = (1 − λsd)R

1 − λs.

With an asset price P = 1, the intermediary’s wealth at date 1 is

y + P (1 − y) = y + (1 − y) = 1.

Because the intermediary’s wealth at date 1 is independent of y , any choice ofy is optimal for the intermediary at date 0. If the intermediary gives λsd tothe early consumers it has 1 − λsd (in present value) for the late consumers.But one unit at t = 1 will buy R units at t = 2, so the intermediary can buy(1 − λsd)R units of consumption at t = 2 and since there are 1 − λs lateconsumers each of them will receive (1−λsd)R/ (1 − λs). The late consumersreceive the same amount in each state if and only if d = 1. In general, the lateconsumer’s consumption varies with the state and the intermediary’s problemis to choose d and y to maximize

λU (d) + 1

2

{(1 − λH )U

((1 − λHd)R

1 − λH

)+ (1 − λL)U

((1 − λLd)R

1 − λL

)},

(6.15)

subject to the incentive constraint c2s ≥ d , which we shall assume is satisfied.Since every choice of y is optimal, the intermediary only has to optimize

with respect to d . The first-order condition for a solution to this problem is

λU ′(d) + 1

2

{(1 − λH )U ′

((1 − λHd)R

1 − λH

) −λHR

1 − λH

+ (1 − λL)U′(

(1 − λLd)R

1 − λL

) −λLR

1 − λL

}

= 0

which simplifies to

U ′(d) = R{

λH

2λU ′(

(1 − λHd)R

1 − λH

)+ λL

2λU ′(

(1 − λLd)R

1 − λL

)}. (6.16)

This is analogous to the usual condition U ′ (c1) = RU ′ (c2), except that theterm in braces on the right hand side is a weighted average of the differentmarginal utilities in each state at date 2.


Example 4 (Continued) If the intermediary cannot achieve the first best,what should it do? To illustrate, we go back to the parameters of Example 4and calculate the explicit solution. The first-order condition (6.16) tells us that

d−6 = 3

{0.6

((1 − 0.6d) 3

0.4

)−6

+ 0.4

((1 − 0.4d) 3

0.6

)−6}

,

which can be solved for d = 1.337. This implies that the consumption of thelate consumers is

c2H = (1 − λHd)R

(1 − λH )= (1 − (0.6) (1.337)) (3)

0.4= 1.485

in the high state and

c2L = (1 − λLd)R

(1 − λL)= (1 − (0.4) (1.337)) (3)

0.6= 2.327.

Notice that although the late consumers do quite well in the low state, this doesnot compensate for the low consumption in the high state. If we calculate theequilibrium expected utility (6.15) explicitly, we get

(0.5)−1

5(1.337)−5 + (0.5)

{(0.4)

−1

5(1.485)−5 + (0.6)

−1

5(2.327)−5

}

= −0.030,

whereas the first best is

(0.5)−1

5(1.428)−5 + (0.5)

−1

5(1.715)−5 = −0.024.

Finally, notice that the incentive constraint c2s ≥ d is satisfied in each state,even thoughwe did not impose it on the solution. This justifies our assumptionthat the incentive constraint is satisfied. Moreover, it shows that it is optimalto have no default in equilibrium.

6.4 CONCLUSION

We can summarize the chapter’s conclusions briefly as follows. As long as wehave complete markets for hedging aggregate risk and intermediaries can use

References 189

complete contingent risk-sharing contracts, the equilibrium in a laisser-faireeconomy will be incentive efficient. If intermediaries are forced by transactioncosts to use incomplete contracts, the equilibriumwill be constrained efficient.In either case, it is wrong to suggest that financial crises constitute a sourceof market failure. A central planner subject to the same informational con-straints or the same transaction costs could not do better than the market. Ifcontracts are complete, there is never any need to default. The intermediarycan achieve the same ends by simply altering the terms of the contract. Incom-plete contracts, on the other hand, distort the choices that an intermediarywould otherwise make; relaxing these constraints by defaulting in some statesof nature allows the intermediary to provide the depositor with superior risksharing and/or higher returns. Whether the intermediary chooses to defaultor not, its choices maximize the welfare of its depositors. Since markets arecomplete, prices give the right signals to intermediaries and guide them tochoose the efficient allocation of risk and investments. It is only when mar-kets are incomplete that we encounter inefficiencies that could in principle becorrected by government regulation and lead to a potential improvement inwelfare. We explore the scope for welfare-improving regulation and the formit takes in the next chapter.

REFERENCES

Allen, F. and D. Gale (2004). “Financial Intermediaries and Markets,”Econometrica 72,1023–1061.

Zame, W. (1993). “Efficiency and the Role of Default when Security Markets areIncomplete,”American Economic Review 83, 1142–1164.

7

Optimal regulation

For the most part, the development of financial regulation has been an empiri-cal process, amatter of trial and error, driven by the exigencies of history ratherthan by formal theory.An episode that illustrates the character of this process istheGreatDepression in theUS.Thefinancial collapse in theUSwaswidespreadand deeply disruptive. It led to substantial changes in the laws regulating thefinancial system,many of which shape our current regulatory framework. TheSEC was established to regulate financial markets. Investment and commercialbanking were segregated by the Glass–Steagall Act (subsequently repealed andreplaced by the Gramm–Leach–Bliley Act of 1999). The Federal Reserve Boardrevised its operating procedures in the light of its failure to prevent the finan-cial collapse. The FDIC and FSLIC were set up to provide deposit insurance tobanks and savings and loan institutions.

Looking back, there is no sign of formal theory guiding these changes.Everyone seems to have agreed the experience of the Great Depression wasterrible; so terrible that it must never be allowed to happen again. Accordingto this mind set, the financial system is fragile and the purpose of prudentialregulation is to prevent financial crisis at all costs.Why does themind set of the1930’s continue to influence thinking about policy? What does policy makingcontinue to be an empirical exercise, with little attention to the role of theory?This empirical procedure is unusual. Indeed, the area of financial regulation issomewhat unique in the extent to which the empirical developments have sofar outstripped theory. In most areas of economics, when regulation becomesan issue, economists have tried to identify some specific market failure thatjustifies the proposed intervention. Sometimes they have gone further andhave derived the optimal form of regulation. This has not been the usualprocedure with financial regulation, however.

The purpose of this chapter is to show how the framework developed inChapter 6 can be used as the basis for analyzing optimal regulation. Thewidespread perception that financial systems are “fragile,” together with manyhistorical episodes of financial instability, has created a presumption that regu-lation is required to prevent costly financial crises. In the previous chapter weargued to the contrary that, under conditions analogous to the assumptions

7.1 Capital Regulation 191

of the fundamental theorems of welfare economics, a laisser-faire equilibriummay be efficient. The occurrence of default and financial collapse in equilib-rium does not necessarily indicate a market failure. If a planner using the samecontracting technology can do no better,we say that equilibrium is constrainedefficient. Unless the authorities have access to a superior technology, interven-tion is not justified when the incidence of financial instability is constrainedefficient.

To provide a justification for regulation of the financial system, we firstneed to identify a source of market failure (constrained inefficiency). Then weneed to identify a practical policy that can remedy or at least ameliorate thatfailure. In this chapter we undertake two policy exercises. First we look at thepotential benefits of regulating capital structure. Then we look at the potentialbenefits of regulating liquidity. In each case, we are interested in determiningwhether a laisser-faire equilibrium is constrained efficient and, if not, what canbe done about it. Our view is that it is not enough merely to show that thereexists a welfare-improving policy. We also need to characterize the policy andshow that it can be implemented. A badly designed intervention could makethings worse. If the welfare-improving policy is too complicated or depends oninformation that is unlikely to be available to the policymaker, such mistakesare likely.

7.1 CAPITAL REGULATION

Capital adequacy requirements are rules that specify aminimum level of capitalthat a bankmustmaintain in relation to its assets. This rulemay take the formofa simple fraction of the assets or amore complicated formula. Capital adequacyrequirements are one of the most important instruments of bank regulation.The first Basel Accord imposed uniform capital adequacy requirements onthe banks of all the signatory countries. A second Basel Accord introducesmore sophisticated methods of determining the appropriate level of capital forbanks, but the idea that banks must be compelled to hold the appropriate levelof capital remains a basic principle of the regulatory system.

These accords provide an example of regulation that is empirically ratherthan theoretically motivated. Practitioners have become experts at the detailsof a highly complex system for which there is no widely agreed rationale basedin economic theory.What is the optimal capital structure?Whatmarket failurenecessitates the imposition of capital adequacy requirements? Why can’t themarket be left to determine the appropriate level of capital? We do not findgood answers to these questions in the theoretical literature.

192 Chapter 7. Optimal Regulation

In the literature on capital adequacy, it is often argued that capital adequacyrequirements are necessary to control the moral hazard problems generatedby the existence of deposit insurance. Deposit insurance was introduced in the1930’s to prevent bank runs or, more generally, financial instability. Becausebanks issue insured debt-like obligations (e.g. bank deposits) they have anincentive to engage in risk-shifting behavior. In other words, the bank has anincentive to make excessively risky investments, because it knows that in theevent of failure the loss is borne by the deposit insurance fund and in theevent of success the bank’s shareholders reap the rewards. The existence ofbank capital reduces the incentive to take risks because, in the event of failure,the shareholders lose their capital. Thus, capital adequacy requirements areindirectly justified by the desire to prevent financial crises. A large literatureinvestigates the effect of capital adequacy requirements on risk taking. Whilethe effect of capital adequacy requirements is usually to decrease risk taking, thereverse is also possible (see, e.g. Kim and Santomero 1988; Furlong and Keeley1989; Gennotte and Pyle 1991; Rochet 1992; and Besanko and Kanatas 1996).

The incentive to take risksmay also be offset by the loss of charter valuewhena firm goes bankrupt (see, e.g. Bhattacharya 1982). This effect will be smallerthe more competitive the structure of the banking market. Keeley (1990) hasprovided evidence that the sharp increase in bank failures in the US in theearly 1980’s was due to increased competition in the banking sector and theassociated fall in charter values.

It appears from our review of the literature that the justification for capitaladequacy requirements is found in the existence of deposit insurance. It couldbe argued that an important question is being begged here: one bad policy(deposit insurance) does not justify another (capital adequacy requirements).Even if it is assumed that deposit insurance prevents financial crises, it is notclear why we should want to reduce the incidence of financial crises, still lesseliminate them altogether. As we demonstrated in Chapter 6, the incidence offinancial crises may be socially optimal in a laisser-faire system. And if not,for example, if financial crises involve deadweight losses, it should be recog-nized that regulation also involves administrative costs and distorts economicdecisions. Any analysis of optimal policy must weigh the costs and benefitsof regulation. This can only be done in a model that explicitly models thepossibility of crises.

Hellman et al. (2000) is an exception in the literature on capital adequacyrequirements. Rather than simply taking the existence of deposit insuranceas given, the authors also examine what happens in the absence of depositinsurance. In the rest of the literature, the rationale for deposit insuranceand in particular its role in preventing financial crises is discussed butnot explicitly modeled. In the absence of explicit modeling of the costs of


financial crises, it is difficult to make a case for the optimality of inter-vention. As a corollary, it is difficult to make a case for capital adequacyrequirements as a means of offsetting the risk taking generated by depositinsurance.

Allen and Gale (2003) argue that, in the absence of a welfare-relevantpecuniary externality, banks will choose the socially optimal capital struc-ture themselves, without government coercion. For a long time, policymakershave taken it as axiomatic that crises are best avoided. By contrast, in Allenand Gale’s framework, a laisser-faire financial system with complete marketsachieves a constrained-efficient allocation of risk and resources. When banksare restricted to using noncontingent deposit contracts, default introduces adegree of contingency that may be desirable from the point of view of opti-mal risk sharing. Far from being best avoided, financial crises can actuallybe necessary in order to achieve constrained efficiency. By contrast, avoidingdefault is costly. It requires either holding a very safe and liquid portfolio (andearning lower returns), or reducing the liquidity promised to the depositorsat the intermediate date. In any case, the bank optimally weighs the costs andbenefits and chooses the efficient level of default in equilibrium.

Our argument is that avoidance of crises should not be taken as axiomatic.If regulation is required to minimize or obviate the costs of financial crises, itneeds to be justified by a microeconomic welfare analysis based on standardassumptions. Furthermore, the form of the intervention should be derivedfrom microeconomic principles. After all, financial institutions and financialmarkets exist to facilitate the efficient allocation of risks and resources. A policythat aims to prevent financial crises has an impact on the normal functioningof the financial system. Any government intervention may impose deadweightcosts by distorting the normal functioning of the financial system. One of theadvantages of a microeconomic analysis of financial crises is that it clarifies thecosts associated with these distortions.

In addition to the incentive function, discussed above, bank capital hasanother main function. This is the risk-sharing function. Capital acts as abuffer that offsets the losses of depositors in the event of a bank failure andallows an orderly liquidation of the bank’s assets, thus avoiding the need todispose of assets at “firesale” prices.

These functions of bank capital explain why shareholders and depositorsshould care about the bank’s capital structure, but they do not explain whygovernments need to regulate capital structure. To the extent that capital struc-ture affects the efficiency of risk sharing or the bank’s incentive to take risk, thecosts andbenefits should be internalized by the bank’s objective function. In theabsence of some sort of externality not taken into account by the banks, thereis no obvious reason why the bank, left to its own devices, should not choose


the (socially) optimal capital structure. In other words, we have not (yet) iden-tified a source of market failure that gives rise to a need for intervention by theregulator.

Incomplete markets provide one possible justification for capital regulation,in the sense that pecuniary externalities have an impact on welfare when mar-kets are incomplete, and in that case regulation of capital (or anything else)can potentially improve welfare. In the following sections, we adapt the modelfrom the previous chapter to show that, when markets are incomplete, thereis a role for bank capital to improve risk sharing and a role for governmentintervention to improve welfare.

7.1.1 Optimal capital structure

As usual there are three dates t = 0, 1, 2 and an all-purpose good that canbe used for consumption or investment. There are two assets, a short assetrepresented by a storage technology that yields one unit at date t + 1 for eachunit invested at date t , and a long asset represented by a constant returns toscale technology that yields R > 1 units of the good at date 2 for each unitinvested at date 0.

There is a continuum of identical investors at date 0 each of whom has anendowment of one unit of the good at date 0 and nothing at future dates. Atdate 1 each consumer learns whether he is an early consumer, who only valuesconsumption at date 1,or a late consumer,whoonly values consumption at date2. The probability that an investor becomes an early consumer is 0 < λ < 1.The investors’ attitudes to risk are represented by a VNM utility function. Ifc is the investor’s consumption, his utility is U (c), where the function U (·)satisfies the usual neoclassical properties.

There are two groups of consumers, group A and group B, and exactlyhalf of the consumers belong to each group. There are two aggregate states ofnature, denoted by (H , L) and (L,H ). Each state is equally likely, that is, eachoccurs with probability 0.5. In state (H , L) the fraction of early consumers ingroup A is λH and the fraction of early consumers in group B is λL , where0 < λL < λH < 1. In state (L,H ) the fractions are reversed. Then the fractionof early consumers in each state is given by

λ = 1

2(λH + λL).

The investors’ attitudes toward risk are represented by a VNM utility func-tion. If an investor consumes c units of the good at the appropriate date, hisutility is U (c), where U (·) satisfies all the usual properties.


All uncertainty is resolved at the beginning of date 1, when the true state isrevealed and each investor learns his type.

In the previous chapterwe assumed thatmarketswere complete. Specifically,we assumed the existence of two Arrow securities at date 0, which allow thetransfer of wealth between states (H , L) and (L,H ), and an asset market onwhich the long asset can be traded at date 1. Here we assume that markets areincomplete. Specifically, we assume that there are noArrow securities, but thereis an asset market at date 1.

The incompleteness of markets would have no effect on the allocation of riskif intermediaries served a representative sample of the population. Instead, weassume that intermediaries draw their customers from either groupA or groupB, but not both. One interpretation of this assumption is that groups A andB correspond to different regions and that intermediaries are restricted by lawto operate in only one region. In any case, the heterogeneity of intermediariesgives rise to gains from risk sharing which cannot be realized because marketsare incomplete.

Apart from the incompleteness of markets, we do not impose any frictionson the model. In particular, intermediaries are allowed to use complete risk-sharing contracts. An intermediary takes a deposit of one unit of the goodfrom each consumer at date 0 and invests it in a portfolio (x , y) consisting of xunits of the long asset and y units of the short asset. In exchange, the consumergets a consumption stream (c1H , c2H , c1L , c2L), where c1H is the consumptionpromised if he withdraws at date 1 when the proportion of early consumersis λH , c2H is the consumption promised if he withdraws at date 2 when theproportion of early consumers is λH , and so on.

In order to discuss capital structure, we introduce a class of risk neutralinvestors to provide capital to intermediaries (see Gale 2003, 2004). Eachinvestor is assumed to have a large endowment of the good at date 0 andnothing at dates 1 and 2. The investors are risk neutral, but their consump-tion must be non-negative (otherwise, the investors could absorb all risk andthe first best would be achieved). Capital is assumed to be expensive in thesense that investors demand a higher return than the intermediary’s invest-ment opportunities can provide. We model the opportunity cost of capital byassuming that investors are impatient: one unit of consumption at date 0 isworth ρ > R units of future consumption. For every unit of capital investedin the intermediary at date 0 the investors will demand an expected return of ρ

units in the future. Since the intermediary’s investments cannot yield a returnhigher than R, the intermediary has to transfer some of the depositors’ returnsto the investors to compensate them for the use of their capital. Even thoughcapital is costly, it is optimal to raise a positive amount of capital because itallows for improved risk sharing.


The intermediary offers investors a contract (e0, eH , eL), where e0 denotesthe amount of capital provided by the investors at date 0 and eH and eL denotethe returns paid to the investors when the fractions of early consumers are λHand λL , respectively. Without loss of generality, we can assume that eH and eLare paid at date 2 because equilibrium requires that the date 1 price of date2 consumption p ≤ 1, so the good is always at least as cheap at date 2 as atdate 1. Investors will supply capital to the bank only if the returns cover theiropportunity cost, that is,

1

2(eH + eL) ≥ ρe0. (7.1)

Since there is a large number of investors, eachof whomhas a large endowment,competition among investors implies that they get no surplus in equilibrium,that is, the inequality (7.1) holds as an equation. So we can assume withoutloss of generality that the intermediary chooses a portfolio (x , y), capital struc-ture (e0, eH , eL), and consumption plan (c1H , c2H , c1L , c2L) to maximize theexpected utility of the typical depositor subject to the investors’ participationconstraint (7.1) and the feasibility constraints. At date 0 the total investmentis constrained by the depositor’s endowment and the capital supplied by theinvestors:

x + y ≤ e0 + 1.

At date 1, the intermediary’s budget constraint is

λs c1s + (1 − λs)pc2s + pes ≤ y + Px

for s = H , L where P = Rp is the price of the asset at date 1.Given there is no aggregate uncertainty the price p will be determined in

the usual way. In order for the banks to be willing to hold both assets betweendates 0 and 1 they must be indifferent between them so

p = 1

R; P = 1.

Since groups A and B are symmetric, and in each state of nature one grouphas a high proportion and one has a low proportion of early consumers, themarket-clearing conditions at date 1 and date 2 are

1

2(λHc1H + λLc1L) = y


and

1

2((1 − λH )c2H + eH + (1 − λL)c2L + eL) = Rx .

From the second of the two budget constraints, we can see that if eH andeL are both positive, then the first-best risk sharing must be achieved, thatis, c2H = c2L . Otherwise, one can increase expected utility by reducing es inone state and increasing it in the other. For example, suppose that c2H < c2L .Then paying the investors eH − ε in state H and eL + ε in state L satisfiesthe investors’ participation constraint (7.1) and makes consumers better offbecause consumption is raised by ε/(1 − λH ) in state H and lowered byε/(1 − λL) in state L so the change in expected utility is proportional to

(1 − λH )U ′(c2H )ε

1 − λH− (1 − λL)U

′(c2L)ε

1 − λL> 0.

The marginal value of insurance is zero when risk sharing is complete whereasthe marginal cost of capital is positive. So it is never optimal to hold enoughcapital to achieve complete risk sharing. Consequently, es must be zero in atleast one state, the one inwhich consumption c2s is lower. For example, supposethat σ > 1. Then we know by the usual argument that c1s > pc2s and averageconsumption is lower when the proportion of early consumers is high. Thenthe optimal capital structure should increase consumption in the high stateand reduce it in the low state. So eH = 0 and 1

2eL = ρe0.

Proposition 1 Suppose consumers have a constant degree of relative riskaversion σ and let (e0, eH , eL) be the optimal capital structure, where e0 > 0.Then

eH > eL = 0

if σ < 1 and

eL > eH = 0

if σ > 1.

In the last chapter we saw that, when there is no aggregate uncertainty,a Pareto-efficient allocation gives every consumer a consumption allocation(c1, c2) that is independent of the state of nature. Whether the efficient allo-cation can be achieved depends on the cost of capital ρ. If ρ is very high, inrelation toR, the optimal capital structure will entail a small infusion of capitale0 at date 0, the intermediary’s ability to smooth consumption between states


will be limited, and the first best will not be attained. It is tempting to con-clude in cases like this that the market has failed because the market outcomeis not Pareto-efficient; but this assumes that the planner is not subject to thetransaction costs and other frictions that prevent markets from being com-plete. Before we decide that the market has failed and that some interventionis required, we should ask whether the central planner could do better if hewere constrained to use only the trading opportunities available to the marketparticipants. For example, it is clear that a planner can improve on the laisser-faire allocation by transferring goods from intermediaries whose depositorshave a low marginal utility of consumption to intermediaries whose depos-itors have a high marginal utility of consumption. In doing so, the planneris performing the function of the missing markets for Arrow securities thatallow intermediaries to transfer wealth across states and achieve the first best.But if the market participants are prevented by transaction costs or other fric-tions from making these trades, perhaps the planner will be too. This suggeststhat the appropriate test for market failure is to ask whether a planner couldimprove on the laisser-faire allocation using the same technology available tothe market participants.

It is not entirely clear what the technology available to the planner should be,but one approach would be to restrict the planner to altering decisions madeat date 0 and requiring him to allow the market to determine the allocationat dates 1 and 2 in the usual way. This would ensure that we are not grantingthe planner a questionable technological advantage over the market. We willsay that a laisser-faire allocation is constrained efficient if the planner cannotimproveon itmerely by changing the allocation at date 0 and leaving themarketto determine the future allocation (Geanakoplos and Polemarchakis 1986).

Since the intermediary chooses an optimal capital structure and an opti-mal investment portfolio and consumption plan, the planner can do betterthan the intermediary only if he changes the equilibrium price. Without achange in price, the choice of the planner is identical to the choice set of theintermediaries. Then it is easy to see that forcing the intermediary to adopt adifferent capital structure cannot improve welfare because with no aggregateuncertainty the equilibrium asset price is determined by the condition that therates of return on the two assets should be equalized in the usual way. In otherwords, the equilibriumprice is independent of the capital structure. Since thereis no pecuniary externality that can be exploited by the regulator, forcing theintermediary to raise more or less capital can only distort the optimal decision.

Proposition 2 Under the maintained assumptions, the laisser-faire equilib-rium is constrained efficient and welfare cannot be improved by changing theequilibrium capital structure.


7.1.2 Models with aggregate uncertainty

The case we have examined is very special. Because the asset price P is aconstant at date 1, it is determined by the requirement that the returns onthe short and long assets be equal. This means that no change in portfoliosor capital structure at date 0 can have any effect on prices at date 1 and, aswe have seen, price changes are the essential ingredient of any improvementin welfare. In models with aggregate uncertainty, the asset price fluctuatesbetween states. Although there is a first-order condition that constrains thedistribution of prices, capital adequacy requirements can have an effect onequilibrium prices and hence have some impact on welfare. Since equilibriawith incomplete markets are typically not constrained efficient, these changesin asset prices can be manipulated to increase welfare. The crucial question,however, is what kind of capital regulation will lead to an improvement inwelfare. It is not obvious that requiring intermediaries to hold more capitalwill be beneficial. In fact, simple examples with no pathological features cangive rise to the surprising conclusion that increasing capital lowers welfare andreducing capital increases welfare.

To illustrate these results, we describe a model studied by Gale and Özgür(2005). The model is identical to the one described above except for thestructure of liquidity shocks. All individuals and intermediaries are ex anteidentical. Ex post there are intermediaries of two types. One type consistsentirely of early consumers and the other type consists of late consumers.There are two (aggregate) states of nature, H and L, which occur with prob-ability 0.5. The proportion of early consumers in state s is denoted by λs , where1 > λH > λL > 0. The proportion of intermediaries consisting of early con-sumers in state s is λs and the probability that any intermediary has only earlyconsumers is λs too. If λi is the proportion of early consumers in intermediaryi then

λi ={

1 w. pr. λs in state s,0 w. pr. 1 − λs in state s,

for s = H , L.The existence of aggregate uncertainty requires some additional complexity

in the intermediaries’ contracts with consumers and investors. Specifically,the payments made to both groups will depend, in general, on both theintermediary’s state (1 or 0) and on the aggregate state (H or L).

A consumer deposits his entire endowment with a single intermediary whooffers him a consumption contract c = {(c1s , c2s) : s = H , L} in exchange,where c1s is consumption offered to an early consumer at date 1 in state s andc2s is the consumption offered to a late consumer at date 2 in state s.


The intermediary writes a contract e = {(e0, e1s , e2s) : s = H , L} withinvestors, where e0 ≥ 0 is the amount of capital invested at date 0, e1s ≥ 0 isthe amount of the good promised to investors in state s if all the depositors areearly consumers and e2s ≥ 0 is the amount of the good promised in state s atdate 2 if all the depositors are late consumers.

In order to reduce the volatility of consumption, the capital structure eshould be chosen so that payments to investors occur when consumption ishigh and not when it is low. Since there are four possible payment oppor-tunities, this leaves a lot of scope for designing the optimal risk sharingarrangements. As usual, because capital is costly, it is not optimal to elim-inate the fluctuations in consumption altogether. This means that changes inprices will have income effects that can increase the ex ante expected utility ofdepositors.

Since the intermediaries are assumed to choose their capital structure opti-mally, taking as given the prices corresponding to each state of nature, it is clearthat capital regulation can improve welfare only by changing the equilibriumprices. The impact of these income effects may be complex. For example, whenan intermediary has only early consumers, it sells its holding of the long asset tomeet its obligations to its depositors. An increase in asset prices will thereforeraise their consumption. For an intermediary that has only late consumers,the effect will be reversed. If late consumers are doing better on average thanearly consumers, the net effect on ex ante utility may be beneficial. But then weneed to consider the possibility that an increase in asset prices in one state maynecessitate a reduction in another. Ultimately, the question is: what change incapital structure will effectively increase welfare?

The answer found by Gale and Özgür depends, not surprisingly, on thedegree of relative risk aversion. They consider a model with constant relativerisk aversion and solve for equilibrium numerically for various parametricassumptions. They find that, if the degree of risk aversion is high enough(greater than σ ≈ 2), a reduction in bank capital reduces price volatility andincreases welfare. For lower risk aversion (i.e. lower than σ ≈ 2), an increase inbank capital increases volatility and welfare. The intuition behind these resultsappears to be that forcing banks to raise costly capital will raise both theirinvestment in the short asset and in the long asset, but it raises investmentin the short asset less than investment in the long asset. This is because thebank is trying to minimize the cost of capital by investing excess capital in thehigher-yielding asset. As a result of this shift in portfolio composition, the assetmarket becomes less liquid and this reduces asset prices in the high state andincreases volatility overall.

It is not known how robust these results are when the model specification isaltered or generalized,but even if the results turnout to be special they reinforce

7.2 Capital Structure with Complete Markets 201

the lesson that,when general-equilibrium effects are involved, it is very difficultto predict the macroeconomic effect of changes in capital structures across thefinancial system. Until we have a general theory to guide us, caution in policymaking would seem to be advisable.

7.2 CAPITAL STRUCTURE WITH COMPLETE MARKETS

The function of bank capital in the preceding section is to allow risk sharingbetween risk neutral investors and risk averse depositors. The investors’ returnsare concentrated in those states where the demand for liquidity is low relativeto the supply. By varying the investors’ returns across states, it is possible toreduce the fluctuations in the depositors’ consumption, in other words, it ispossible to provide depositors with insurance against liquidity shocks or assetreturn shocks. An optimal capital structure is one way to provide this kindof insurance, but it is not the only way. If markets at date 0 were complete,insurance could be provided through the markets and the need for capitalwould be eliminated entirely.

Let the good at date 0 be the numeraire and let pts denote the price of oneunit of the good at date t = 1, 2 in state s = H , L. An intermediary will wantliquidity at date 1 if its depositors are early consumers and at date 2 if they arelate consumers. So what the intermediary wants is an option on the good ateach date.What will this option cost? Since the probability of having depositorswho are early consumers is λs in state s, the cost of the option should be λsp1sfor date 1 and (1 − λs)p2s for date 2. If the intermediary uses the completemarkets to obtain an optimal risk-sharing contract for its depositors, it willoffer a consumption plan (c1H , c2H , c1L , c2L) to maximize the expected utility

1

2{λHU (c1H ) + (1 − λH )U (c2H )} + 1

2{λLU (c1L) + (1 − λL)U (c2L)}


λHp1Hc1H + (1 − λH )p2Hc2H + λLp1Lc1L + (1 − λL)p2Lc2L ≤ 1.

Similarly, investors can usemarkets to spread their consumption over the threedates. As usual, there is no loss of generality in assuming that they consumeonly in the first and last periods, so they will choose a bundle (e0, eH , eL) wheree0 is the amount of the good supplied at date 0 and es is the consumption at


date 2 in state s, to maximize

1

2{eH + eL} − ρe0 (7.2)


p2HeH + p2LeL ≤ e0. (7.3)

Because markets are complete, investments in the short and long assets yieldzero profits. It does not matter who makes the investments, since they addnothing to wealth or the possibility of risk sharing, so we can, without loss ofgenerality, assume that all investments are made by a representative firm. Inequilibrium, the firm will buy the goods supplied by the intermediaries andthe investors at date 0, that is, 1 + e0, invest them in the short and long assets,and use the returns from these assets to supply goods to the investors and theintermediary at dates 1 and 2. In addition to the zero-profit conditions, theusual market-clearing conditions must be satisfied (see Chapter 6).

Because the usual assumptions, including the assumption of complete mar-kets, are satisfied, the fundamental theorems of welfare economics ensure thatan equilibrium is Pareto-efficient (or incentive efficient if the incentive con-straints are binding). Thus,with completemarkets,we get efficient risk sharingbetween investors, on the one hand, and the intermediaries and their cus-tomers, on the other. Although it is intermediated by the market, the provisionof insurance is similar to what happens in a model with capital structure.The investors supply “capital” at date 0 that must be invested in real assetsin order to provide future consumption. They take their returns in the formof consumption at date 2. Because they are risk neutral, they will consumeonly in the state where the price p2s is a minimum, leaving more consump-tion for the depositors in other states. This bunching of consumption by theinvestors in a single state allows the depositors to smooth their consumptionacross states and, in particular, to consume more in the state with a high costof consumption.

The existence of complete markets not only provides a perfect substitutefor optimal capital structure, thus making capital redundant, it also makes theoptimal capital structure indeterminate. This is because any capital structurecan be undone by transactions in the market. Suppose that (e0, eH , eL) is anaction that maximizes (7.2) subject to (7.3). Because the objective functionand the constraint are both linear in (e0, eH , eL), the optimal trade must satisfy

1

2

{eH + eL

}− ρ e0 = 0

7.2 Capital Structure with Complete Markets 203

and

p2H eH + p2LeL − e0 = 0.

Now suppose that the intermediary adopts (e0, eH , eL) as its capital structure.Because the intermediary has access to complete markets, the only effect ofcapital structure is on the intermediaries budget constraint. This capital struc-ture is optimal for the intermediary in the sense that it minimizes the costsubject to the investors’ participation constraint. Because the cost is zero, itdoes not affect the set of consumption plans the intermediary can afford andthe optimal plan (c1H , c2H , c1L , c2L) will satisfy the budget constraint

λHp1Hc1H + (1 − λH )p2Hc2H + λLp1Lc1L

+ (1 − λL)p2Lc2L + p2H eH + p2LeL − e0 ≤ 1.

Finally, if both (e0, eH , eL) and (e0, eH , eL) maximize the investors’ objec-tive function (7.2) subject to the budget constraint (7.3), then so does(e0, eH , eL) − (e0, eH , eL), so we can assume that the intermediaries chooseto trade (e0, eH , eL) − (e0, eH , eL) in equilibrium. Then the combined effect ofthe optimal contract between the intermediaries and the investors (e0, eH , eL)and the net trade in themarkets (e0, eH , eL)−(e0, eH , eL) is precisely equivalentto the trade (e0, eH , eL) in the original equilibrium.

Thus, the optimal capital structure is indeterminate. This is simply a versionof the Modigliani–Miller theorem.

In the analysis of the preceding section, there are two sources of liquidityfor intermediaries that suffer a bad liquidity shock at date 1. One is the capitalstructure negotiated with investors, which allows payment to investors to bereduced in the event of a bad liquidity shock; the other is asset sales to otherintermediaries. The capital structure is chosen optimally by the intermediaries,so this is not a source of inefficiency. Rather the incompleteness of marketsforces banks to sell assets at prices that are determined ex post by the demandfor and supply of liquidity in each state. As we have seen, the ex post provisionof liquiditymay be inefficient because intermediaries end up selling their assetsat a low price in states where the marginal utility of their depositors is high,the opposite of what good insurance requires. By contrast, when markets arecomplete at date 0, the intermediary can transfer wealth across states at theprevailing prices and this ensures that the marginal rates of substitution acrossstates are equalized for all depositors.

It is worth noting that the heterogeneity of intermediaries ex post is crucialfor this result. If intermediaries were identical at date 1 there would be no gainsfrom trade and hence no need for markets. To put it another way, markets


are always effectively complete in a Robinson Crusoe economy. So marketscan be incomplete without any effect on efficiency if each intermediary has arepresentative sample of consumers at each date. The indeterminacy of capitalstructure does not survive, however: if there are no markets for contingentcommodities the capital structure is determinate because it is only through thecapital structure that efficient risk sharing can be achieved.

7.3 REGULATING LIQUIDITY

Now we turn to the study of liquidity regulation, that is, the possibility ofimproving welfare by regulating the amount of the short asset held in equilib-rium. To keep things simple, we eliminate the risk neutral investors, so there isno provision of capital. Otherwise, the assumptions are the same as in Section7.1. The example is based on one in Allen and Gale (2004).

Apart from the incompleteness of markets, we do not impose any frictionson the model. In particular, intermediaries are allowed to use complete risk-sharing contracts. An intermediary takes a deposit of one unit of the goodfrom each consumer at date 0 and invests it in a portfolio (x , y) consisting of xunits of the long asset and y units of the short asset. In exchange, the consumergets a consumption stream (c1H , c2H , c1L , c2L), where c1H is the consumptionpromised if he withdraws at date 1 when the proportion of early consumersis λH , c2H is the consumption promised if he withdraws at date 2 when theproportion of early consumers is λH , and so on.

Because there is no aggregate uncertainty, we can assume the price of theasset at date 1 is independent of the state, that is,

PHL = PLH = P .Since equilibrium requires that both assets are held at date 0, the one-periodholding returns on both assets must also be the same at date 0. The return onthe short asset is equal to one and the return on the long asset is equal to P , sothe equilibrium price must be

P = 1.

If P = 1 is the price of R units of the good at date 2, the price of one unit isp = 1/R.

Because of the symmetry of the model, we focus on a symmetric equilib-rium and describe the behavior of a representative intermediary. Althoughintermediaries are heterogeneous, they solve essentially the same decision

7.3 Regulating Liquidity 205

problem. Each has an equal probability of having a high or a low propor-tion of early consumers, and it is the number of early consumers, λH or λL ,that matters to the intermediary, not the state of nature (H , L) or (L,H ). Sowe can describe the intermediaries’ decision problem in terms of the inter-mediary’s “state” H or L, meaning the number of early consumers is λH orλL .

At date 0, an intermediary takes in a deposit of one unit from each con-sumer and invests it in a portfolio (x , y) and a contingent consumption planc = (c1H , c2H , c1L , c2L). Since all intermediaries choose the same portfolioand consumption plan, we can describe an allocation by a triple (x , y , c). Anallocation (x , y , c) is attainable if it satisfies the market-clearing conditions ateach date. At date 0 this requires that total investment equal the endowmentof goods:

x + y = 1.

At date 1, the supply of goods is y (since P < R no one will want to invest inthe short asset at this date). The demand for goods will be 1

2 (λHc1H + λLc1L)since half the intermediaries are in state H and half are in state L. Thenmarket-clearing at date 1 requires

1

2(λHc1H + λLc1L) = y .

Similarly, market-clearing at date 2 requires

1

2[(1 − λH )c2H + (1 − λL)c2L] = Rx .

Each intermediary chooses the portfolio (x , y) and the consumption plan cto maximize the expected utility of the typical depositor. At the equilibriumasset price P = 1, all portfolios (x , y) will have the same value at date 1:

Px + y = x + y = 1

so the intermediary is indifferent among all feasible portfolios.Moreover, sincethe value of the intermediary’s portfolio is the same in each state, the intermedi-ary can maximize expected utility in each state independently by choosing(c1s , c2s) to maximize

λsU (c1s) + (1 − λs)U (c2s)



λs c1s + (1 − λs)c2sR

= 1.

So an equilibrium consists of the equilibrium asset price P = 1 and an attain-able allocation (x , y , c) such that, for each state s = H , L, the consumptionplan (c1s , c2s) maximizes expected utility subject to the budget constraint inthat state.

7.3.1 Comparative statics

Before we can begin the analysis of optimal regulation, we have to establish anumber of comparative static properties.We begin by focusing on the behaviorof an intermediary in a particular state. Since the two states are symmetric, wecan suppress the reference to the state for the time being. If the fraction of earlyconsumers is λ at date 1 then the intermediary’s budget constraint at date 1 is

λc1 + (1 − λ)c2R

= 1. (7.4)

Maximizing expected utility λU (c1) + (1 − λ)U (c2) subject to this budgetconstraint gives the usual first-order condition

U ′(c1) = RU ′(c2). (7.5)

It is important to notice that, although the number of early consumers doesnot appear in the first-order condition, it does affect the optimal consumptionallocationbecause it appears in thebudget constraint. Thefirst-order conditionimplies that c1 and c2 vary together, so a change in λ will typically raise orlower consumption at both dates. In fact, an increase in λ increases the lefthand side of (7.4) if and only if c1 > c2/R. Intuitively, if the present value ofearly consumption is greater than the present value of late consumption, anincrease in the proportion of late consumers will increase the present value oftotal consumption. In order to satisfy its budget constraint, the intermediarywill have to reduce average consumption. If c1 < c2/R, an increase in λ willhave the opposite effect.

Proposition 3 For each date t = 1, 2, consumption is lower in state H thanin state L if c1 > Rc2 for all pairs (c1, c2) satisfying the first-order condition(7.5). Conversely, consumption is higher in state H than in state L if c1 < Rc2for all order pairs satisfying the first-order condition (7.5).


This effect turns out to be crucial for the comparative static properties of theequilibrium, so we investigate it in more detail. To do this, consider the specialcase where the VNM utility function exhibits constant relative risk aversion σ :

U (c) = 1

1 − σc1−σ .

The first-order condition (7.5) becomes

(c1)−σ = R(c2)

−σ ,

which implies

c1 = c2R −1σ =( c2R

)R1− 1

σ .

Then it is easy to see that

c1 >c2R

⇐⇒ R1− 1σ > 1 ⇐⇒ σ > 1.

The present value of consumption is higher at date 1 than at date 2 if and onlyif the degree of relative risk aversion is greater than one.

Proposition 4 If the consumers’ degree of relative risk aversion is a constantσ , then c1 > Rc2 for all pairs (c1, c2) satisfying the first-order condition (7.5)if and only if σ > 1.

The role of risk aversion in determining the consumption allocation has anintuitive interpretation. The first-order condition implies that marginal util-ity is lower at date 2 than at date 1. In other words, c1 is less than c2. Otherthings being equal (i.e. holding constant the expected value of consumption)a risk averse consumer would prefer to reduce the uncertainty about his levelof consumption. Other things are not equal, of course. In order to reduceconsumption-risk it is necessary to hold more of the short asset and less of thelong asset. A more liquid portfolio yields lower average returns and provideslower average consumption. Given this trade-off between consumption riskand average consumption levels, we should expect the degree of risk aversionto affect the intermediary’s choice. The more risk averse the consumer, themore he values insurance and the lower the average level of consumption he iswilling to accept in order to smooth consumption over the two periods. Thecritical value σ = 1 corresponds to the case in which it is optimal to equalizethe present value of consumption between the two periods: c1 = c2/R. If riskaversion is less than one, the high returns from delaying consumption out-weigh the value of insurance and the optimal consumption allocation gives


early consumption a lower present value than late consumption. If risk aver-sion is greater than one, the value of insurance outweighs the return fromdelaying consumption and the optimal consumption allocation gives earlyconsumption higher present value than late consumption.

Figure 7.1 illustrates the relationship between risk aversion and consump-tion risk. When the degree of relative risk aversion is very low, it is optimal totake a high risk of being an early consumer who gets very low consumptionin order to have a chance of being a late consumer who gets a very high con-sumption. By contrast, when the degree of relative risk aversion is high, thedifference between consumption at the two dates is reduced to the point wherethe present value of future consumption is lower than the value of presentconsumption.

4321

2.5

2

1.5

1

0.5

0

–0.5s

Rc 1

/c2

Figure 7.1. Illustration of relationship between σ and the ratio of the present valuesc1 and c2/R for R = 2.

The correspondence between risk aversion and the present value of con-sumption at the two dates translates immediately into a correspondencebetween risk aversion and the slope of the consumption functions relatingthe value of λ and the level of consumption. If σ is greater than one, bothc1 and c2 decline as λ increases. If σ is less than one, c1 and c2 increase as λ

increases.

Proposition 5 Suppose the consumers’ degree of relative risk aversion isa constant σ and let c = (c1H , c2H , c1L , c2L) be the optimal consumption


allocation chosen by the representative intermediary. Then c1H < c1L andc2H < c2L if and only if σ > 1.

Figure 7.2 illustrates the optimal levels of consumption at date 1 and date 2as a function of λ, the fraction of early consumers, when the degree of relativerisk aversion is greater than one. For each value of λ, consumption at date 1 isless than consumption at date 2, but the ratio of c2 to c1 is less than R = 2, sothe present value of consumption at date 1 is greater than the present value ofconsumption at date 2. Thus, as λ increases, consumption at both dates falls.

10.750.50.25

2.5

2

1.5

1

0.5

0

–0.5

–1

λ

c 1,c

2

Figure 7.2. Illustration of relationship between λ and (c1, c2) for σ = 2, R = 2.

In terms of welfare, if σ > 1, consumers are better off when the number ofearly consumers is low and, if σ < 1, consumers are better off at both dateswhen the number of early consumers is high. This property of equilibrium isthe key to understanding the welfare effects of any intervention in the market.

7.3.2 Too much or too little liquidity?

In the last chapter we saw that,when there is no aggregate uncertainty, a Pareto-efficient allocation gives every consumer a consumption allocation (c1, c2) thatis independent of the state of nature. The preceding analysis shows that, in theabsence of Arrow securities that allow wealth to be transferred across states,the equilibrium consumption allocation will depend on the fraction of early


consumers and hence on the state. So an unconstrained central planner cancertainly achieve a higher level of welfare than the intermediaries can in theabsence of Arrow securities; however, as we argued in Section 7.1, the relevantquestion is whether the equilibrium is constrained efficient.

It is well known that models with incomplete markets are typically notconstrained efficient, so our presumption is that by manipulating decisionsat date 0 the planner can potentially improve on the laisser-faire allocation.More precisely, there exists a welfare-improving intervention, but it is notobvious what it is. This is an important distinction for the policymaker: it isnot sufficient to know that there exists some (possibly complex) policy thatwill improve welfare. The policymaker needs to know what to do; otherwise,he may make things worse. Our objective in this section is to characterize thewelfare-improving policies. As we shall see, even in the context of this simpleexample, it is not easy to say what the right policy is.

In what follows, we assume that the planner’s intervention is limited to con-trolling the portfolio choices of the intermediaries, specifically, the amount ofthe short asset they hold. The planner’s ability to improve on the laisser-faireallocation depends upon the possibility of changing the equilibrium prices.In a laisser-faire equilibrium, the intermediaries choose portfolios and con-sumption plans optimally, taking as given the asset prices they face in thefuture. It is rational for intermediaries to treat prices as parameters beyondtheir control because the number of intermediaries is so large that no singleintermediary can have a significant impact on the market-clearing asset price.A planner, on the other hand, does not take future prices as given. Althoughintermediaries will make future consumption decisions taking prices as given,the planner anticipates that his influence over the intermediaries’ portfoliodecisions will have some impact on prices. It is through the effect of portfoliochoices on prices that the planner can potentially improve the welfare of theintermediaries’ depositors.

Suppose that the planner requires intermediaries to hold more of the shortasset in their portfolios. This action will have two immediate effects. First, itwill have a direct effect on portfolios, increasing y and reducing x . Second, itwill change the market-clearing asset prices at date 1. Presumably, increasingthe amount of liquidity and reducing the stock of the long asset will increasethe asset price. The consumers’ welfare depends only on consumption, so theeffect of any change in portfolios and prices on welfare will be indirect. Thechange in theportfolio and the asset prices at date 1will shift the intermediaries’budget constraints at date 1, causing them to choose different consumptionplans. We know that all feasible portfolios (x , y) have the same market valueat date 1 because the long and short assets have equal returns. Thus, to afirst approximation, a small change in the portfolio has no impact on the


intermediary’s budget constraint. A change in prices, on the other hand, willhave an impact on the intermediary’s budget constraint, in fact, it will haveboth substitution and income effects. In analyzing the impact of the policyintervention on consumers’ welfare, we only need to pay attention to incomeeffects. The envelope theorem assures us that, since the intermediary choosesthe consumption plan to maximize the typical consumer’s expected utility, asmall movement along the budget constraint will have no impact on expectedutility. So it is only the income effect of the price change that is relevant forwelfare analysis.

Suppose that a change in y increases P (and p) in each state. What willbe the income effect of this change? Consider the budget constraint of theintermediary in state H ,

λHc1 + (1 − λH )pc2H = y + pRy .An increase in p increases the left hand because it increases the present value ofthe consumption provided at date 2. An increase in p also increases the righthand side because it increases the present value of the return to the long asset.Thus, the income effect of an increase in p is given by Ry− (1−λH )c2H . Moreprecisely, this is the amount by which we could increase expenditure on theearly consumers and still balance the budget in state H . Similarly, the incomeeffect of the price change in state L is given by Ry − (1 − λL)c2L .

Now the market-clearing condition at date 2 requires that

1

2[(1 − λH )c2H + (1 − λL)c2L] = Ry ,

so, in this case, the income effects in the two states sum to zero

Ry − (1 − λH )c2H + Ry − (1 − λL)c2L = 0.

The income effect of a price change raises consumption in one state andlowers consumption by an equal amount in the other state. When markets arecomplete, so that marginal utility of consumption is the same in each state, atransfer of consumption fromone state to another has no effect.Whenmarketsare incomplete, by contrast, the marginal utility of consumption is typicallyhigher in one state than in the other and this makes it possible for incomeeffects to increase expected utility.

Suppose, for example, that the degree of relative risk aversion is greater thanone, so that consumption at each date is higher in state L than in stateH . Thisimplies that

Ry − (1 − λH )c2H > 0 > Ry − (1 − λL)c2L ,


so the income effect of an increase in p is positive in state H and negative instate L. Furthermore, the marginal utility of consumption for early consumersis higher in state H than in state L. Thus, if we assume that the change in realincome is reflected in the consumption of early consumers in each state, thegain to the early consumers in stateH will more than offset the loss to the earlyconsumers in state L. Formally,

U ′(c1H ){Ry − (1 − λH )c2H

}> −U ′(c1L)

{Ry − (1 − λL)c2L

}.

By a similar argument, we would get the opposite result if we started byassuming that the degree of relative risk aversion was less than one.

There is no loss of generality in assuming that only the consumption of theearly consumers changes. By the envelope theorem, we cannot do better bydividing the change in consumption between early and late consumers. Thus,we have a necessary and sufficient condition for an improvement in welfarefrom an increase in p which we state as the next proposition.

Proposition 6 Starting at the laisser-faire equilibrium, an increase in p (or P)is welfare improving (i.e. will increase depositors’ expected utility) if and onlyif the degree of relative risk aversion is greater than one.

It remains to connect the change in price at date 1 to the change in portfoliosat date 0. Intuitively, we expect an increase in y to be associated with anincrease in P (or p) because it increases the supply of the good at date 1 anddecreases the supply of the long asset. However, in the laisser-faire equilibriumintermediaries are indifferent between the two assets at date 0 and the quantitythey hold in their portfolios is determined by the requirements for marketclearing at date 1. If consumption at date 1 goes up, the amount of the shortassetmust increase to provide that consumption. So whether an increase in p isassociated with an increase in y depends on the reaction of the intermediary’sconsumption plans.What we can say is that for any small change in p there willbe an equilibrium in which intermediaries’ portfolio choices are constrainedappropriately.

Irwin et al. (2006) have extended the simple example for considering theregulation of liquidity considered in Allen and Gale (2004) and this chapter.They show that while minimum liquidity controls can lead to Pareto improve-ments when intermediaries are homogeneous ex ante, this is not the casewhen they are heterogeneous ex ante. In this case other policies such as state-contingent tax and transfer schemes or state-contingent lender of last resortpolicies are necessary for improvements in efficiency.

What lessons can we draw from the kind of exercise conducted in thissection? For starters, a policymaker who wishes to increase welfare needs to


have detailed knowledge about the risk-sharing arrangements undertaken bythe financial sector. More generally, the effects of increased liquidity in themarket are indirect and work through the general-equilibrium determinationof asset prices and consumption plans. So it takes a lot of information aboutthe structure of the model and the equilibrium to predict the effect of policyon equilibrium. If this is true in a fairly trivial example, one would expect theproblems facing a policymaker in the ‘real’ world to be quite challenging.


There have been a number of good surveys and overviews of banking regula-tion. These include Herring and Santomero (2000), Santos (2001), Freixas andSantomero (2004), Barth et al. (2006). For this reason, this section will be keptshort.

There is widespread agreement that the most important rationale for bank-ing regulation is the prevention of systemic risk. However, as discussed initiallyin this chapter there is not agreement about the nature of the market failurethat leads to this systemic risk. The policies that have been used to try and limitsystemic risk include capital adequacy ratios, liquidity requirements, reserverequirements, deposit insurance, and asset restrictions. Another importantmotivation for regulation is consumer protection. Conflict of interest rulesand interest rate ceilings on loans are examples of policies aimed at protectingconsumers. Other policies, such as competition policy, are directed at industrygenerally but enhance the efficiency of the banking system. The governmentalso tries to implement broader social objectives, such as the prevention ofmoney laundering through reporting requirements for large cash transactions.Dewatripont and Tirole (1994) have pointed to another category of rationalefor justifying banking regulation. Bankers, like the managers of any othercoporation, need to be monitored by investors. Bank depositors are particu-larly unsuited for this role because they typically have limited resources andlimited experience. Regulation can be a substitute for monitoring to ensurethe bank acts in the interests of the depositors.


In this chapter, we have argued that the first step in finding optimal regula-tory policies is the identification of market failure(s). The model considered inChapter 6 provides conditions under which market forces lead to an efficient


allocation of resources. Moreover, the optimal allocation can involve financialcrises. So it is not the case that eliminating systemic risk is always optimal. Acareful analysis of the costs and benefits of crises is necessary to understandwhen intervention is necessary. This analysis is typically missing from pro-posals for capital adequacy regulations such as the Basel Accords. Incompletefinancial markets provide one plausible source of market failure and a possiblejustification for capital regulation. The form this regulation takes is complexand informationally intensive, however, so it is not clear that this provides thebasis for a practical policy. One also needs to keep in mind the continuingfinancial innovation that allows banks to hedge risks in ever more sophisti-cated ways. Whether this makes the assumption of complete markets realisticis an open and important empirical question. Allen and Gale (2006) containsa further discussion of some of these issues.

Regulation is only one of the ways that governments intervene in the finan-cial system. The other important way is through the actions of the centralbank. The next chapter considers the role of monetary policy.

REFERENCES

Allen, F. and D. Gale (2003). “Capital Adequacy Regulation: In Search of a Rationale,”in Economics for an Imperfect World: Essays in Honor of Joseph Stiglitz edited byR. Arnott, B. Greenwald, R. Kanbur and B. Nalebuff, Cambridge, MA: MIT Press,83–109.

Allen, F. and D. Gale (2004). “Financial Intermediaries and Markets,”Econometrica 72,1023–1061.

Allen, F. and D. Gale (2006). “Systemic Risk and Regulation,” in R. Stulz and M. Carey(eds.), Financial Risk and Regulation. Cambridge, MA: NBER..

Barth, J., G. Caprio Jr. and R. Levine (2006). Rethinking Banking Regulation: Till AngelsGovern, Cambridge, New York and Sydney: Cambridge University Press.

Besanko, D. and G. Kanatas (1996). “The Regulation of Bank Capital: Do CapitalStandards Promote Bank Safety?” Journal of Financial Intermediation 5, 160–183.

Bhattacharya, S. (1982).“Aspects of Monetary andBanking Theory andMoralHazard,”Journal of Finance 37, 371–384.

Dewatripont, M. and J. Tirole (1994). The Prudential Regulation of Banks, Cambridge,MA: MIT Press.

Freixas, X. and A. Santomero (2004). “Regulation of Financial Intermediaries: A Dis-cussion,” in Credit Intermediation and the Macroeconomy, Models and Perspectives,edited by S. Bhattacharya, A. Boot and A. Thakor, Oxford and New York: OxfordUniversity Press.

Furlong, F. and M. Keeley (1989). “Capital Regulation and Bank Risk-Taking: A Note,”Journal of Banking and Finance 13, 883–891.

References 215

Gale, D. (2003). “Financial Regulation in a Changing Environment,” in T. Courcheneand E. Neave (eds.), Framing Financial Structure in an Information Environment.Kingston,Ontario: JohnDeutsch Institute for the Study of Economic Policy,Queen’sUniversity.

Gale, D. (2004). “Notes on Optimal Capital Regulation,” in P. St-Amant and C.Wilkins(eds.), The Evolving Financial System and Public Policy. Ottawa: Bank of Canada.

Gale,D. andO.Özgür (2005).“AreBankCapital RatiosTooHighorTooLow:RiskAver-sion, Incomplete Markets, and Optimal Capital Structures,” Journal of the EuropeanEconomic Association 3, 690–700.

Geanakoplos, J. and H. Polemarchakis (1986). “Existence, Regularity, and ConstrainedSuboptimality of Competitive AllocationsWhen the Asset Market Is Incomplete,” inW. Heller, R. Starr, and D. Starrett (eds.), Essays in honor of Kenneth J. Arrow: Volume3, Uncertainty, information, and communication. Cambridge, New York and Sydney:Cambridge University Press, 65–95.

Gennotte, G. and D. Pyle (1991). “Capital Controls and Bank Risk,” Journal of Bankingand Finance 15, 805–824.

Hellmann, T., K. Murdock, and J. Stiglitz (2000). “Liberalization, Moral Hazard inBanking, and Prudential Regulation: Are Capital Requirements Enough?”AmericanEconomic Review 90, 147–165.

Herring, R. and A. Santomero (2000). “What is Optimal Financial Regulation?” TheNew Financial Architecture, Banking Regulation in the 21st Century, edited by B. Gup,Westport, Connecticut: Quorum Books, 51–84.

Irwin, G., V. Saporta, and M. Tanaka (2006). “Optimal Policies to Mitigate FinancialStability when Intermediaries are Heterogeneous,”working paper, Bank of England.

Keeley, M. (1990). “Deposit Insurance, Risk, and Market Power in Banking,”AmericanEconomic Review 80, 1183–1200.

Kim, D. and A. Santomero (1988). “Risk in Banking and Capital Regulation,” Journalof Finance 43, 1219–1233.

Rochet, J-C. (1992). “Capital Requirements and the Behaviour of Commercial Banks,”European Economic Review 36, 1137–1178.

Santos, J. (2001). “Bank Capital Regulation in Contemporary Banking Theory: AReview of the Literature,” Financial Markets, Institutions and Instruments 14,289–328.

8

Money and prices

In the preceding chapters,wehave assumed that intermediaries offer depositors“real” contracts, that is, contracts that are denominated in terms of goods. Forexample, an intermediary offers its depositors a deposit contract that promisesc1 units of the good if the depositor withdraws at date 1 and c2 units of thegood if the depositor withdraws at date 2. In practice, the terms of depositcontracts and debt contracts in general are written in “nominal” terms, thatis, they specify payments in terms of money. Economists often justify thesubstitution of “real” for“nominal”contracts by claiming thatmoney is a“veil”that hides the reality we are interested in, namely, the goods and services thatfirms produce and individuals consume; but the fact that contracts are writtenin terms of money has some important implications that are not capturedby “real” models. The most important feature of nominal contracts is thatchanges in the price level (the general level of prices measured in terms ofmoney) change the real value of the contract.

A. C. Pigou was one of the first to point out the impact of the price level onthe real value of debt. More precisely, outside money, the part of the moneysupply that constitutes a claimon the government, represents part of the privatesector’s net wealth. A fall in the price level by definition increases the real valueof outside money and, consequently, increases the private sector’s real wealth,that is, its wealth measured in terms of goods and services. This “wealth effect”subsequently came to be known as the “Pigou effect.” Pigou used it to criticizeKeynes’ argument that a general fall in prices would have no effect on demandfor goods and services. Keynes relied on the familiar homogeneity property ofdemand functions: because demand and supply depend only on relative prices,an equal proportionate change in prices and wages should have no effect ondemand and supply. Pigou argued to the contrary that a fall in the general levelof prices would increase perceived wealth and that this might lead consumersto demand more goods and services.

An important element of Pigou’s argument was the distinction betweeninside and outside money. Outside money consists of currency and depositswith the Federal Reserve System, sometimes called the monetary base or high-poweredmoney. Insidemoney consists of bank deposits and other liabilities of

Chapter 8. Money and Prices 217

the banking system. Inside money consists of obligations of the private sectorto itself whereas outside money represents an obligation of the governmentto the private sector. A fall in the price level will increase the value of privatedebts, thus reducing the net wealth of the debtor; but there will be an equaland opposite effect on the wealth of the creditor. When aggregated over theentire private sector, these debts and credits will cancel out, thus leading toa zero effect on the net wealth of the private sector as a whole. This leavesthe wealth effect of price changes on outside money but since the quantity ofoutside money is usually small this wealth effect is also small.

To a first approximation, thismay be a reasonable approach in normal times,but in cases of financial crises where the change in the price level is large, as itwas during the Great Depression of the 1930’s, it can be misleading. The rea-son is that a very large fall in the price level may make it impossible for firmsand individuals to pay their debts, in which case they may be forced to defaultand seek the protection of bankruptcy. If bankruptcy had no real effect on thevalue of the creditors’ claims, it might not matter to Pigou’s argument; but inpractice bankruptcy often involves large deadweight costs. These include notonly the legal costs of liquidation but also the loss of organizational capitalthat occurs when an enterprise ceases to be a going concern and the disloca-tion that can spread throughout the economy while productive activities arebeing reorganized. These deadweight losses, which can amount to a significantfraction of GDP, are one of the reasons why financial crises are regarded withsuch horror by policymakers.

Just as a fall in the general price level has the effect of increasing the levelof real indebtedness in the economy, an increase in the price level has theeffect of reducing the real level of indebtedness. Often governments burdenedwith a large national debt have resorted to the expedient of reducing it bycreating inflation. This is often referred to as “monetizing” the debt, since theinflationary process begins with an increase in the money supply and the finaleffect is to reduce the real value of the debt and to increase the quantity ofmoney balances held by individuals. It does not require hyperinflation to havea significant effect on the real value of the national debt. Steady but modestinflation had a very significant impact on the real value of the US nationaldebt left at the end of the Second World War, for example, and many othercountries had a similar experience.

More recent episodes also illustrate the importance of nominal debt con-tracts.A good example is theAsian crisis of 1997.Manyof the countries affectedby that crisis had large amounts of external debt denominated in terms of dol-lars. Foreign lenders were unwilling to accept debt issued in terms of the localcurrency because they did not trust the government to maintain its value. Bycontrast, the same lenders felt somewhat protected if their investment was

218 Chapter 8. Money and Prices

denominated in dollars. One example of a country affected by the existence offoreign currency debt was Thailand. Many Thai firms had borrowed in dollarsto make investments in the Thai economy. The value of the Thai currency, thebaht, was pegged to the dollar, which may have made the loans seem less riskyto the borrowers. However, when the dollar began to rise in terms of other cur-rencies, the baht had to rise with it, thus making Thai exports more expensiveand causing the balance of trade to deteriorate. Speculators anticipated thatthe government would have to devalue the baht and began to sell. The currencyattack led to a balance of payments crisis and the baht was devalued. This leftthe Thai businesses that had borrowed large amounts of dollars in a tenuousposition. Their revenues were denominated in baht and their debts in dollars.The debts had grown relative to their ability to pay and many had to default.

Another illustration of the importance of nominal contracts comes fromJapan, which suffered from low investment and slow growth throughout the1990’s as a result of the bursting of an asset-price bubble in 1990. The gov-ernment of Japan adopted Keynesian remedies (government expenditure onconstruction projects and low interest rates) with little effect. At one point,there was a considerable fear that deflation (a general fall in the prices ofgoods and services) might cause a serious problem. Since many Japanese firmswere heavily indebted and many banks were saddled with non-performingloans, this fear of deflation was quite real. Fortunately, the deflation was nottoo severe and this episode passed without further mishap, but it provided anobject lesson on the relevance of the price level when debt is denominated interms of money.

In this chapter we want to focus on some of the more benign aspects of theprice level when debt is denominated in nominal terms, in particular, we showhow variations in the price level, by varying the real value of debt, can intro-duce a desirable level of contingency in risk-sharing contracts. Simple debtcontracts promise fixed repayments, independently of the state of nature. Anoptimal risk-sharing contract, on the other hand,will typicallymake paymentscontingent on the state. By varying prices, a contract that is fixed in nominalterms can be made contingent in real terms. This increase in the contingencyof the contract may improve risk sharing under certain conditions.

8.1 AN EXAMPLE

We illustrate the role of price level variability in supporting efficient risk sharingby presenting a simple example based on one in Allen and Gale (1998). Asusual, we assume that time is divided into three periods or dates, indexed by

8.1 An Example 219

t = 0, 1, 2. There is a single good, which can be used for consumption orinvestment, at each date. There are two (real) assets, a short asset, representedby a storage technology which produces one unit of the good at date t + 1for every unit invested at date t , and a long asset, represented by a long-terminvestment technology which produces R > 1 units of the good at date 2 forevery unit invested at date 0. There is a large number of identical consumers atdate 0, eachwith an endowment of one unit of the good at date 0 andnothing atdate 1. The consumers are subject to liquidity preference shocks: they are eitherearly consumerswho only value consumption at date 1 or late consumerswhoonly value consumption at date 2. There are two states of nature, s = H , L,with probabilities πH and πL , respectively. The probability that an individualbecomes an early consumer depends on the state. Let λs denote the fraction ofearly consumers, which equals the probability of becoming an early consumer,in state s = H , L. We assumed that 0 < λL < λH < 1. All uncertainty isresolved at the beginning date 1, when the true state of nature is revealed andeach consumer learns whether he is an early or a late consumer.

We assume that free entry and competition force intermediaries tomaximizethe expected utility of the typical depositor, subject to a zero profit constraint.The intermediary takes a deposit of one unit from each depositor at date 0and invests it in a portfolio (x , y) consisting of x units of the long asset andy units of the short asset. In exchange, the intermediary offers a risk-sharingcontract c = (c1H , c2H , c1L , c2L) that promises cts units of consumption toa consumer who withdraws at date t = 1, 2 in state s = H , L. If U (c) denotesa consumer’s utility from consuming c units of the good, the expected utilityof this contract is ∑

t ,s

πs {λsU (c1s) + (1 − λs)U (c2s)} (8.1)

and a competitive intermediary will choose the portfolio (x , y) and the con-sumption allocation c to maximize (8.1) subject to the feasibility constraints

x + y ≤ 1; (8.2)

λs c1s ≤ y , ∀s = H , L; (8.3)

λs c1s + (1 − λs) c2s ≤ y + Rx , ∀s = H , L. (8.4)

Suppose that the portfolio (x , y) has been chosen and consider the optimalallocation of consumption in a given state s. The consumption allocation(c1s , c2s) has to maximize expected utility in that state, that is, maximize

λsU (c1s) + (1 − λs)U (c2s)


subject to the feasibility conditions (8.3) and (8.4). We know that c1s must beless than or equal to c2s ; otherwise we could increase expected utility by usingthe short asset to shift consumption to date 2. We also know that if λs c1s < y ,so that the short asset is actually being used to transfer consumption betweenthe dates, then c1s must equal c2s ; otherwise we could increase expected utilityby reducing the investment in the short asset and shifting consumption to date1. Thus, there are two situations to consider:

λs c1s < y and c1s = c2s = y + Rxor

c1s = y

λs≤ c2s = Rx

1 − λs.

For a given portfolio (x , y) we can see that the consumption allocation (x , y)is a function of λ as illustrated in Figure 8.1.

c

Rx /(1 – λ)

y/λ

0 λ* λ

y + Rx

Figure 8.1. Consumption c1 and c2 at dates 1 and 2, respectively, as functions of theproportion λ of early consumers.

What can we say about the equilibrium consumption allocation? Since λL <

λH we know that there are three possible situations, depending on where λLand λH stand in relation to λ∗. One of these possibilities is illustrated inFigure 8.2 and this case can be ruled out immediately.

If both λH and λL are to the left of λ∗ as illustrated in Figure 8.2, then thelevel of consumption is the same at both dates and in both states. But this isinconsistent with the first-order condition that can be derived in the usual wayfrom the portfolio choice problem. By reducing the investment in the shortasset and increasing the investment in the long asset, the intermediary can

8.1 An Example 221

c

Rx/(1 – λ)

y/λ

λL λH


c

Rx/(1 – λ)

y/λ

λL λH


reduce consumption by 1/λs at date 1 in both states and increase consumptionbyR/(1−λs) at date 2 in both states. This will leave expected utility unchangedif and only if U ′(c1s) = RU ′(c2s), which is impossible if c1s = c2s . So the caseillustrated in Figure 8.2 cannot arise. This leaves us with the case illustrated inFigure 8.3 or the case illustrated in Figure 8.4.

In either of the cases illustrated in Figures 8.3 and 8.4 the level of consump-tiondepends on the state of nature at both date 1 anddate 2. This is problematic


c

Rx/(1 – λ)

y/λ

λL λH


for the intermediary if it is restricted, for informational reasons or because oftransaction costs, to using simple debt contracts, that is, deposit contractsthat promise a fixed amount of consumption at each date. A deposit contractcannot reproduce the consumption patterns shown in Figures 8.3 and 8.4.

Default can introduce some additional contingency, and under very specialconditions it may allow the intermediary to achieve the first-best allocation,but it may also introduce deadweight costs that lead to a loss of welfare.

Here is where a nominal contract may have advantages. Suppose the inter-mediary promises to pay the depositor D units of money either at date 1 orat date 2, depending when he chooses to withdraw. If the price level (i.e. theprice of the good in terms of money) is denoted by pts at date t = 1, 2 in states = H , L, then the consumers’ consumption must satisfy

p1Hc1H = p2Hc2H = p1Lc1L = p2Lc2L = D.

If the central bank can control the price level, it can ensure that the optimalconsumption allocation is achieved simply by regulating the price level so thatthe real value of the deposit D/pts equals the optimal consumption allocationin each state at each date.

Notice that the fact that c1s ≤ c2s implies that p1s ≥ p2s for each state s.Declining nominal prices simply implies a positive real interest rate. An alter-native but equivalent approach would assume that the bank pays nominalinterest on accounts, so that the early withdrawers receive D and the late with-drawers receive (1+ r)D. This would be consistent with stable or rising prices,but would leave real values unchanged.

8.2 Optimal Currency Crises 223

Given the right level of prices and a fixed nominal payment, the consumersmust get the right level of consumption, but is this an equilibrium? What isgoing on behind these formulae? In order to pay its depositors at date 1, theintermediary has to borrow money from the central bank. It pays out thismoney to the depositors who demand repayment, and they in turn spend thismoney on goods. The banks supply goods in exchange for money and earnjust enough to repay their loan to the central bank at the end of the period.The same procedure is followed at date 2. Again, we need to distinguish caseswhere the early consumers consume the entire returns to the short asset fromthe case where there is excess liquidity and some of the short asset is rolledover to the last date. In the first case, we have c1s < c2s and p1s > p2s . Thenominal return to holding the short asset until date 2 is p2s − p1s < 0, so thebank should be willing to sell all of the goods it produces from the short assetand this is what happens in equilibrium since λs c1s = y . On the other hand, ifλs c1s < y , then p1s = p2s and the return on the short asset is p2s−p1s = 0 andthe intermediary is content to hold back some of the goods it has produced atdate 1 and reinvest them in the short asset. In either case, the market clears andthe banks are doing the best they can taking the prices as given. The circularflow of income is illustrated in Figure 8.5.

Banks

Households

D Dc

Figure 8.5. Banks pay deposits D to households which use the money to purchaseconsumption c .

8.2 OPTIMAL CURRENCY CRISES

Just as a change in the domestic price level changes the real value of nominalcontracts within a country, a change in the country’s exchange rate changes the


external value of debt denominated in the domestic currency. In this sectionwe will develop a simple version of the model in Allen and Gale (2000) andshow that appropriate adjustments in the exchange rate play a role in achievingoptimal risk sharing by transferring risk from a small country to the rest of theworld.

Imagine a country that is so small in relation to the rest of the world (ROW)that, for all practical purposes, we can assume that what happens in the smallcountry has no impact on the ROW. The small country faces risk in the formof uncertainty about the size of its GDP. We assume that the rest of the world(ROW) is risk neutral and we normalize the gross return on a riskless assetto one. Optimal risk sharing between the small country and the ROW wouldrequire all risk to be borne by the risk neutral party, that is, the ROW. If thesmall country’s output were sold on the world equity market, its value wouldbe equal to the expected value E[Y ] of its GDP. Because of imperfections incapital markets, the ROW may not be willing to take an equity stake in thesmall country. If debt is used to finance investment in the small country andeither (a) the debt is denominated in terms of foreign currency or (b) theexchange rate is fixed, the risk of fluctuations in GDP will be borne by thedomestic investors. By contrast, if the country can issue debt denominated inthe domestic currency and adjust its exchange rate appropriately, we shall seethat most of the risk can be transferred to the ROW.

Domestic investors have an endowmentW which they want to invest. Thecountry’s output is given by a production function

Y = θF(K ),

where Y is output, K is the domestic capital stock, θ (a random variable) isa productivity shock and F(·) is an increasing, concave function. Suppose itis possible for the domestic banking system to borrow from the ROW at thecompetitive rate of interest and lend to domestic entrepreneurs who investin the domestic capital stock. In equilibrium, the marginal product of capitalshould equal the opportunity cost of funds, that is, the equilibrium capitalstock K ∗ should satisfy

E[θF ′(K ∗)] = 1.

The banking sector is assumed to invest in a portfolio consisting of risk-less international bonds and loans to domestic producers. Domestic investorsdeposit their endowment W in the banking sector. The banking sector buysB international bonds and lends K ∗ to domestic producers. The demand forforeign borrowing to finance investment in domestic loans and international

8.2 Optimal Currency Crises 225

Table 8.1. Banking sector balance sheet.

Assets Liabilities

Loans K ∗ Domestic deposits WSecurities B Foreign deposits B + K ∗ −WTotal B + K ∗ Total B + K ∗

bonds is B+K ∗ −W . Table 8.1 shows the banking sector’s assets and liabilitiesin real terms.

We assume that the price level in the ROW is constant and equal to one, sothe international currency (dollars) is equivalent to goods and the exchangerate e, which is the dollar price of one unit of the domestic currency, is equalto the real value of the domestic currency. Now suppose the domestic banksissue debt (deposits) equal to D units of the domestic currency. Part of this isbought by domestic investors in exchange for their endowment W and partby foreigners in exchange for dollars and international bonds B. The futureexchange rate will adjust so that the real value of the debt is equal to thecombined value of the country’s output and the stock of foreign assets:

eD = Y + B.

Let k be the fraction of the debt held by the ROW. Since the ROW is riskneutral, its expected return must equal the return on the safe asset, so its totalexpected return equals the investment it made in the small country. Thus,

kE[Y + B] = B + K ∗ −W ,

or

k = B + K ∗ −WE[Y + B] .

Then the residual amount held by domestic investors is given by

(1 − k) (Y + B) =(1 − B + K ∗ −W

E[Y + B]) (Y + B)

= (E [Y ]− (K ∗ −W )) Y + BE[Y + B] .


The term Y+BE[Y+B] is random, but as B → ∞ this term converges in probability

to 1 and this proves that the domestic investors share converges to a constant:

limB→∞(1 − k) (Y + B) = E [Y ]− (K ∗ −W ) .

In other words, by borrowing a large amount and using it to form a largeportfolio consisting of a small fraction of domestic loans and a large fractionof international bonds, the banking system can export most of the risk to theROW, thus improving the welfare of risk averse domestic investors at no cost.

The role of exchange rate fluctuations in this exercise is limited to convertingdomestically denominated debt into an equity stake.Without this assumption,the risk would still be borne by domestic investors.

8.3 DOLLARIZATION AND INCENTIVES

In the preceding section we assumed that foreign investors were willing toaccept debt denominated in the domestic currency and showed that thisarrangement was consistent with optimal risk sharing. Two very importantassumptions underlie this result. The first is that foreign investors correctlyanticipate fluctuations in the future exchange rate and adjust the value of thedebt accordingly. The second is that the domestic government can committo an exchange rate policy. One of the reasons why foreign investors may beunwilling to accept debt denominated in the domestic currency is precisely thefear that the government will inflate the currency or reduce the exchange ratein order to expropriate the foreign investors. For this reason, foreign investorsmay insist that debt be denominated in an international currency such as thedollar. Alternatively, the government of the small country may take steps toensure that the exchange rate will bemaintained, for example, by establishing acurrency board that pegs the exchange rate to a foreign currency and removesmonetary policy from the control of the government.

By giving up control of the exchange rate and monetary policy, the govern-ment may be paying a high price to obtain foreign investment, but it has beenargued that the discipline imposed by such arrangements may pay dividendsto the country. A simple example based on Gale and Vives (2002) will illus-trate the idea. Suppose that a representative entrepreneur wants to undertakea risky venture using foreign capital. The venture requires an investment of Kand produces a revenue of YH if successful and YL if unsuccessful, where

YH > K > YL > 0.

8.3 Dollarization and Incentives 227

The probability of success depends on the effort taken by the entrepreneur. Ifthe entrepreneur takes effort the probability of success is π > 0; otherwise itis zero. The cost of effort is C .

Suppose that the entrepreneur finances his project by issuing real or dollarbonds with face value D. Foreign investors are assumed to be risk neutral andthe return on the safe asset is one, so the foreign investors are willing to lend Kif and only if the expected repayment is equal to K . If the entrepreneur takesno effort, the output will be YL with probability one, so the most the investorswill receive is YL which is less than K . So the investors will not be willing tobuy the bonds issued by the entrepreneur unless they are sure that he will makean effort.

Now suppose that the entrepreneur makes an effort. If the project is suc-cessful, he can repayD < YH but if the project fails he can only repay YL < D.Thus, the total repayment is

πD + (1 − π)YL = K ,

which implies that πD = K − (1 − π)YL . In the event of success, theentrepreneur receives a profit of YH − D; in the event of failure he receivesnothing. Thus, his expected profit is

π(YH − D) = πYH + (1 − π)YL − K .

If effort is costly, the profit he receives may not be sufficient to encourage theentrepreneur to undertake the effort required to make the project successfulwith probability π . If the cost of effort is C then

πYH + (1 − π)YL − K < C .

Suppose next that he gets a private benefit B from the success of the project,for example, he develops a reputation as a successful entrepreneur that allowshim to take advantage of future profitable opportunities. Then if

πYH + (1 − π)YL − K + πB > C

the entrepreneur is willing to undertake the project with costly effort andeveryone is happy.

Now suppose that the entrepreneur borrows in the domestic currency andthe exchange rate e is controlled by the government. Ex ante, the governmenthas an incentive to say that it will maintain the exchange rate in order toencourage foreign investment. Ex post, if the investment projects are unsuc-cessful, it has an incentive to reduce the exchange rate so that the face value


of the debt is only eD = YL . This allows the entrepreneur to retain his privatebenefit B from avoiding default and still allows the foreign investors to retaintheir claim to the output YL in the low state. It might be thought that sincethe foreign investors receive the same payment in each state, they do not carewhether the domestic currency is devalued or not. They should care becausethe entrepreneur’s incentives have changed. Since he receives the private bene-fit B in any event, it no longer affects his willingness to take effort. The net gainto taking effort is now

πYH + (1 − π)YL − K < C ,

so the entrepreneur does not take effort, the outcome is YL for sure, and theforeign investors are unwilling to finance the project.


Money and banking crises

Most models of banking crises, such as those discussed in Chapter 3, do notconsider the role of money. Banks contract with depositors in real terms. Allenand Gale (1998) showed how the use of nominal deposit contracts and theinjection of money by a central bank could prevent crises. As discussed above,variations in the price level allowed risk to be shared and acted as a substitutefor state contingent contracts.

Smith (2002) considers a model where spatial separation and limited com-munication introduces a role for money into a standard banking model withearly and late consumers. He shows that the lower the inflation rate and nom-inal interest rate, the lower is the probability of a banking crisis. Reducing theinflation rate to zero in line with the Friedman rule eliminates banking crises.However, this is inefficient as it leads banks to hold excessive cash reserves atthe expense of investment in higher yielding assets.

In addition to the literature on money and banking crises, there is also aliterature on how the cost of bail-outs after banking crises should be funded.Should they be paid for using tax proceeds or by money creation? Boyd et al.(2004) show that, in a general equilibrium context where savings, deposits,bank reserves, and the inflation tax base are endogenous, monetizing at leastpart of the cost can be desirable.

Diamond and Rajan (2001) develop a model where banks have special skillsto ensure that loans are repaid. By issuing real demand deposits, banks canprecommit to recoup their loans. This allows long-term projects to be funded


and depositors to consume when they have liquidity needs. However, thisarrangement leads to the possibility of a liquidity shortage in which bankscurtail credit when there is a real shock. Diamond and Rajan (2006) introducemoney and nominal deposit contracts into the model to investigate whethermonetary policy can help alleviate this problem. They assume there are twosources of value for money. The first arises from the fact that money canbe used to pay taxes (the fiscal value). The second is that money facilitatestransactions (the transactions demand). They show that the use of moneycan improve risk sharing since price adjustments introduce a form of statecontingency to contracts. However, this is not the only possibility. In somecases variations in the transaction value of money can lead to bank failures.Monetary intervention can help ease this problem. If the central bank buysbonds with money, this changes liquidity conditions in the market and allowsbanks to fundmore long-termprojects thanwould be possible in the absence ofintervention. Themodel thus provides a different perspective on the operationof monetary policy through bank lending.

Currency crises and twin crises

There is a large literature on currency crises. Flood and Marion (1999) providea survey. Krugman (2000) contains a number of analyses of historic and recentcurrency crises. Fourçans and Franck (2003) is an excellent book on thesubject. Chui and Gai (2005) explains the global games approach to analyzingcrises. This literature review will therefore be kept brief.

The first-generation currency crisis models were designed to explain theproblems experienced by a number of Latin American countries in the 1970’sand early 1980’s. An important characteristic of these episodes was that theyhad their origins in macroeconomic imbalances. The classic references hereare Krugman (1979) and Flood and Garber (1984). These papers show how afixed exchange rate plus a government budget deficit leads to a currency crisis.In equilibrium, there cannot be a discontinuous change in the exchange rate asthis would lead to an arbitrage opportunity. Instead, the exchange rate adjustscontinuously so that the real rate of return on domestic currency is equatedto the real rate of return on foreign currency. The fiscal deficit is covered bya combination of depletion of foreign reserves and an inflation tax on thedomestic money stock. When the exchange rate hits the level that would occurwithout support there is a speculative attack and reserves are exhausted.

Although the first-generationmodels hadmany nice features, they had diffi-culty explaining episodes such as the Exchange Rate Mechanism (ERM) crisisof 1992, in which the pound and the lira dropped out of the mechanism.First, the timing of these currency crises is very unpredictable. Second, there


are often “discontinuous” jumps in exchange rates. Finally, the models assumethat no steps are taken by the government to eliminate deficits.

These problems led to the development of second generation models. Forexample, Obstfeld (1996) shows how a conditional government policy canlead to multiple equilibria – one without a speculative attack and one with aspeculative attack. The existence of multiple equilibria and uncertainty aboutthe timing of an attack permit a discontinuous jump in the exchange rate. Theoutcome of the attack depends on the resources the government is willing tocommit to maintain the exchange rate.

Equilibrium selection is an important issue in this literature. Morris andShin (1998) show how asymmetric information can lead to uniqueness ofequilibrium in models of currency crises as coordination games. Chui and Gai(2005) provide an excellent account of the so-called global games approach.

The large movements in exchange rates that occurred in many East Asiancountries in 1997 led to the development of a third generation of currencycrisismodels. In contrast to thefirst and second generationmodels,manyof thecountries that experienced problems in the recent EastAsian crisis had pursuedconsistent and sustainable macroeconomic policies. This characteristic of therecent crises prompted a re-examination of theoretical models of currencycrises.

Another characteristic of the South East Asian crises is the simultaneouscrises that occurred in the banking systems of these countries. Kaminsky andReinhart (1999) have investigated the relationship between banking crises andcurrency crises. They find that in the 1970’s, when financial systems werehighly regulated in many countries, currency crises were not accompanied bybanking crises. However, after the financial liberalization that occurred duringthe 1980’s, currency crises and banking crises became intertwined. The usualsequence of events is that problems in the banking sector are followed by acurrency crisis and this in turn exacerbates and deepens the banking crisis.Although banking crises typically precede currency crises, the common causeof both is usually a fall in asset values due to a recession or a weak economy.Often the fall is part of a boom–bust cycle that follows financial liberalization.It appears to be rare that banking and currency crises occur when economicfundamentals are sound.

In recent episodes, despite the apparent inter-relationship between currencycrises and banking crises, the literatures on these topics have for the mostpart developed separately. Important exceptions are Chang andVelasco (2000,2001). The first paper develops a model of currency and banking crises basedon the Diamond and Dybvig (1983) model of bank runs. Chang and Velascointroduce money as an argument in the utility function. A central bank con-trols the ratio of currency to consumption. Different exchange rate regimes


correspond to different rules for regulating the currency-consumption ratio.There is no aggregate uncertainty in thesemodels: banking and currency crisesare “sunspot” phenomena. In other words, there are at least two equilibria, a“good” equilibrium in which early consumers receive the proceeds from short-term assets and late consumers receive the proceeds from long-term assets anda “bad” equilibrium in which everybody believes a crisis will occur and thesebeliefs are self-fulfilling. Chang and Velasco (2000) shows that the existence ofthe bad equilibrium depends on the exchange rate regime in force. In someregimes, only the good equilibrium exists; in other regimes there exists a badequilibrium in addition to the good equilibrium. The selection of the good orthe bad equilibrium is not modeled. In Chang and Velasco (2001) a similarmodel is used to analyze recent crises in emerging markets. Again, there is noaggregate uncertainty and crises are sunspot phenomena.

Corsetti et al. (1999) have developed a model of twin crises designed toexplain the Asian meltdown in 1997. The basic reason that twin crises occur intheir framework is because of moral hazard arising from government guaran-tees. Foreigners are willing to lend for unprofitable projects against the promiseof future government bailouts. When the project payoffs turn out to be lowthere will be a banking crisis. The prospect of the government using seignior-age to finance the bailouts leads to the prospect of inflation and so the currencyalso collapses.

Kaminsky and Reinhart’s (1999) finding that crises are related to economicfundamentals is consistent with work on US financial crises in the nineteenthand early twentieth centuries. Gorton (1988) andCalomiris andGorton (1991)argue that the evidence is consistent with the hypothesis that banking crisesare an essential part of the business cycle rather than a sunspot phenomenon.As discussed above, Allen and Gale (2000) extends the model of Allen andGale (1998) to consider twin crises. A model is developed in which the “twin”crises result from low asset returns. Large movements in exchange rates aredesirable to the extent that they allow better risk sharing between a country’sbank depositors and the international bond market.

Dollarization

De Nicoló et al. (2003) point out that the domestic use of foreign currency, inotherwords dollarization,has increased substantially in recent years.Accordingto Galindo and Leiderman (2005), this has been particularly true in LatinAmerica. Some countries have adopted the use of foreign currency entirely butmost have adopted a mixed system where dollars (or euros) are used alongsidethe domestic currency.


What are the benefits of dollarization? Dollarization began in most coun-tries as a reaction to high inflation rates and a way to protect the value ofsavings. De Nicoló et al. (2003) find that dollarization leads to more financialintermediation only if inflation rates are already high. The theory of Gale andVives (2002), discussed above, suggests that another benefit of full dollarizationis the disciplining of firms.

The potential problems associated with dollarization are more varied.Although it is often argued that dollarization limits the abilities of governmentsto act independently,Reinhart et al. (2003) donot find significant differences inthe ability of governments in partially dollarized economies to control inflationor stabilize output. They also find that revenue from seigniorage is generallyindependent of the level of dollarization. Citing the example of Peru, Galindoand Leiderman (2005) argue that partial dollarization does not prevent coun-tries from running an independent monetary policy. However, dollarizationdoes appear to make countries more vulnerable to adverse shocks.


Most of this book is concerned with theories of crises where contracts are inreal terms. This chapter has considered the effect of allowing contracts to be innominal terms andmoney to be introduced into the analysis. Three effects havebeen focused on. The first is that variations in the price level allow nominaldebt to become effectively state contingent so that risk sharing is improved.The second is that fluctuations in the exchange rate, combined with foreignholdings of domestic debt and domestic holdings of foreign debt, allow riskto be transferred away from the domestic economy to diversified internationalinvestors. Finally, complete dollarizationmay provide good incentives to firms.This is only a small subset of the topics related to the interaction of monetarypolicy and financial crises. Much work remains to be done in this area. Thenext chapter considers a way in which expansive monetary policy can lead tofinancial crises through the creation of bubbles.

REFERENCES

Allen, F. and D. Gale (1998). “Optimal Financial Crises,” Journal of Finance 53, 1245–1284.

Allen, F. and D. Gale (2000). “Optimal Currency Crises,”Carnegie-Rochester ConferenceSeries on Public Policy 53, 177–230.

References 233

Boyd, J., C. Chang, and B. Smith (2004). “Deposit Insurance and Bank Regulationin a Monetary Economy: A General Equilibrium Exposition,” Economic Theory 24,741–767.

Calomiris, C. and G. Gorton (1991). “The Origins of Banking Panics, Models, Facts,and Bank Regulation,” in R. Hubbard (ed.), Financial Markets and Financial Crises,Chicago, IL: University of Chicago Press.

Chang, R. and A. Velasco (2000). “Financial Fragility and the Exchange Rate Regime,”Journal of Economic Theory 92, 1–34.

Chang, R. and A. Velasco (2001). “A Model of Financial Crises in Emerging Markets,”Quarterly Journal of Economics 116, 489–517.

Chui, M. and P. Gai (2005). Private Sector Involvement and International FinancialCrises, Oxford and NewYork: Oxford University Press.

Corsetti, G., P. Pesenti, and N. Roubini (1999). “Paper Tigers? A Model of the AsianCrisis,” European Economic Review 43, 1211–1236.

De Nicoló, G., P. Honohan, and I. Ize (2003). “Dollarization of the Banking System:Good or Bad?”World Bank Policy Research Working Paper 3116,Washington, DC.


Diamond, D. and R. Rajan (2001). “Liquidity Risk, Liquidity Creation and FinancialFragility: A Theory of Banking,” Journal of Political Economy 109, 287–327.

Diamond,D. andR. Rajan (2006).“Money in aTheory of Banking,”American EconomicReview 96, 30–53.

Flood, R. and P. Garber (1984). “Gold Monetization and Gold Discipline,” Journal ofPolitical Economy 92, 90–107.

Flood, R. and N. Marion (1999). “Perspectives on the Recent Currency CrisisLiterature,” International Journal of Finance & Economics 4, 1–26.

Fourçans, A. and R. Franck (2003). Currency Crises: A Theoretical and EmpiricalPerspective, Cheltenham, UK; Northampton, MA, USA: Edward Elgar.

Gale, D. and X. Vives (2002). “Dollarization, Bailouts, and the Stability of the BankingSystem,”Quarterly Journal of Economics 117, 467–502.

Galindo, A. and L. Leiderman (2005). “Living with Dollarization and the Routeto Dedollarization,” Inter-American Development Bank, Research Department,Working Paper #526.

Gorton, G. (1988). “Banking Panics and Business Cycles, Oxford Economic Papers 40,751–781.

Kaminsky, G. and C. Reinhart (1999). “The Twin Crises: The Causes of Banking andBalance-of-Payments Problems,”American Economic Review 89, 473–500.

Krugman, P. (1979). “A Model of Balance of Payments Crises,” Journal of Money, Creditand Banking 11, 311–325.

Krugman, P. (ed.) (2000). Currency Crises, National Bureau of Economic Research,Chicago: The University of Chicago Press.

Morris, S. and H. Shin (1998). “Unique Equilibrium in a Model of Self-FulfillingCurrency Attacks,”American Economic Review 88, 587–597.


Obstfeld,M. (1996).“Models of CurrencyCriseswith Self-fulfilling Features,”EuropeanEconomic Review 40, 1037–1047.

Reinhart, C., K. Rogoff, andM. Savastano (2003)“Addicted to Dollars,”NBERWorkingPaper 10015.

Smith, B. (2002). “Monetary Policy, Banking Crises, and the Friedman Rule,”AmericanEconomic Review 92, 128–134.

9

Bubbles and crises

In the previous chapter we considered the role of money and the price levelin sharing risk. In this chapter we consider the role of money and creditin the determination of asset prices and the prevention of crises. The ideathat the amount of money and credit available is an important factor in thedetermination of asset prices is not new. In his description of historic bubblesKindleberger (1978, p. 54) emphasizes the role of this factor: “Speculativemanias gather speed through expansion of money and credit or perhaps, insome cases, get started because of an initial expansion of money and credit.”

In many recent cases where asset prices have risen and then collapsed dra-matically an expansion in credit following financial liberalization appears tohave been an important factor. Perhaps the best known example of this type ofphenomenon is the dramatic rise in real estate and stock prices that occurredin Japan in the late 1980’s and their subsequent collapse in 1990. Financialliberalization throughout the 1980’s and the desire to support the US dollarin the latter part of the decade led to an expansion in credit. During most ofthe 1980’s asset prices rose steadily, eventually reaching very high levels. Forexample, the Nikkei 225 index was around 10,000 in 1985. On December 19,1989 it reached a peak of 38,916. A new Governor of the Bank of Japan, lessconcerned with supporting the US dollar and more concerned with fightinginflation, tightened monetary policy and this led to a sharp increase in interestrates in early 1990 (see Frankel 1993; Tschoegl 1993). The bubble burst. TheNikkei 225 fell sharply during the first part of the year and by October 1, 1990it had sunk to 20,222. Real estate prices followed a similar pattern. The nextfew years were marked by defaults and retrenchment in the financial system.The real economy was adversely affected by the aftermath of the bubble andgrowth rates during the 1990’s were typically slightly positive or negative, incontrast to most of the post-war period when they were much higher.

Similar events occurred in Norway, Finland, and Sweden in the 1980’s (seeHeiskanen 1993;Drees andPazarbasioglu 1995; Englund andVihriälä 2006). InNorway the ratio of bank loans to nominal GDP went from 40 percent in 1984to 68 percent in 1988. Asset prices soared while investment and consumptionalso increased significantly. The collapse in oil prices helped burst the bubble

236 Chapter 9. Bubbles and Crises

and caused the most severe banking crisis and recession since the war. In Fin-land an expansionary budget in 1987 resulted in massive credit expansion. Theratio of bank loans to nominal GDP increased from 55 percent in 1984 to 90percent in 1990. Housing prices rose by a total of 68 percent in 1987 and 1988.In 1989 the central bank increased interest rates and imposed reserve require-ments to moderate credit expansion. In 1990 and 1991 the economic situationwas exacerbated by a fall in trade with the Soviet Union. Asset prices collapsed,banks had to be supported by the government and GDP shrank by 7 percent.In Sweden a steady credit expansion through the late 1980’s led to a propertyboom. In the fall of 1990 credit was tightened and interest rates rose. In 1991a number of banks had severe difficulties because of lending based on inflatedasset values. The government had to intervene and a severe recession followed.

Mexico provides a dramatic illustration of an emerging economy affected bythis type of problem. In the early 1990’s the banks were privatized and a finan-cial liberalization occurred. Perhaps most significantly, reserve requirementswere eliminated. Mishkin (1997) documents how bank credit to private non-financial enterprises went from a level of around 10 percent of GDP in the late1980’s to 40 percent of GDP in 1994. The stockmarket rose significantly duringthe early 1990’s. In 1994 the Colosio assassination and the uprising in Chiapastriggered the collapse of the bubble. The prices of stocks and other assets felland banking and foreign exchange crises occurred. These were followed by asevere recession.

These examples suggest a relationship between the occurrence of significantrises in asset prices or positive bubbles and monetary and credit policy. Theyalso illustrate that the collapse in the bubble can lead to severe problemsbecause the fall in asset prices leads to strains on the banking sector. Banksholding real estate and stocks with falling prices (or with loans to the ownersof these assets) often come under severe pressure from withdrawals becausetheir liabilities are fixed. This forces them to call in loans and liquidate theirassets which in turn appears to exacerbate the problemof falling asset prices. Inother words there may be negative asset price bubbles as well as positive ones.These negative bubbles where asset prices fall too far can be very damaging tothe banking system. This can make the problems in the real economy moresevere than they need have been. In addition to the role of monetary andcredit policy in causing positive price bubbles there is also the question ofwhether monetary policy has a role to play in preventing asset prices fromfalling too far. In the Scandinavian and Mexican examples discussed above,asset prices quickly rebounded and the spillovers to the real economy wererelatively short-lived. In Japan asset prices did not rebound for a long timeand the real economy has been much less robust. It was only in 2005 that theeconomy started to grow strongly again.

9.1 Agency Problems and Positive Bubbles 237

Despite the apparent empirical importance of the relationship betweenmonetary policy and asset price bubbles there is no widely agreed theoryof what underlies these relationships. This chapter considers the relationshipbetween asset price bubbles, financial crises and the role of the central bank.Section 9.1 looks at the relationship between credit expansion and positivebubbles. Allen and Gale (2000) provide a theory of this based on the exist-ence of an agency problem. Many investors in real estate and stock marketsobtain their investment funds from external sources. If the ultimate providersof funds are unable to observe the characteristics of the investment, there is aclassic risk-shifting problem. Risk shifting increases the return to investmentin risky assets and causes investors to bid up prices above their fundamentalvalues. A crucial determinant of asset prices is thus the amount of credit thatis provided. Financial liberalization, by expanding the volume of credit andcreating uncertainty about the future path of credit expansion, can interactwith the agency problem and lead to a bubble in asset prices.

When the bubble bursts either because returns are low or because the centralbank tightens credit, banks are put under severe strain. Many of their liabilitiesare fixed while their assets fall in value. Depositors and other claimants maydecide to withdraw their funds in anticipation of problems to come. This willforce banks to liquidate some of their assets and this may result in a furtherfall in asset bubbles because of a lack of liquidity in the market. Section 9.2considers how such negative bubbles arise. Rather than focusing on the rela-tionship between the bank and borrowers who make investment decisions asin Section 9.1, the focus is on depositors and their decisions. It is shown thatwhen there is a market for risky assets then their price is determined by “cash-in-the-market pricing” in some states and can fall below their fundamentalvalue. This leads to an inefficient allocation of resources. The central bankcan eliminate this inefficiency by an appropriate injection of liquidity into themarket.

Finally, Section 9.3 contains concluding remarks.

9.1 AGENCY PROBLEMS AND POSITIVE BUBBLES

How can the positive bubbles and ensuing crashes in Japan, Scandinavia, andMexico mentioned above be understood? The typical sequence of events insuch crises is as follows.

There is initially a financial liberalization of some sort and this leads to alarge expansion in credit. Bank lending increases by a significant amount.Some of this lending finances new investment but much of it is used to


buy assets in fixed supply such as real estate and stocks. Since the supplyof these assets is fixed the prices rise above their “fundamentals.” Practicalproblems in short selling such assets prevent the prices from being bid downas standard theory suggests. The process continues until there is some realevent that means returns on the assets will be low in the future. Anotherpossibility is that the central bank is forced to restrict credit because offears of “overheating” and inflation. The result of one or both of theseevents is that the prices of real estate and stocks collapse. A banking crisisresults because assets valued at “bubble” prices were used as collateral. Theremay be a foreign exchange crisis as investors pull out their funds and thecentral bank chooses between trying to ease the banking crisis or protectthe exchange rate. The crises spill over to the real economy and there is arecession.

In the popular press and academic papers, these bubbles and crises are oftenrelated to the particular features of the country involved. However, the fact thata similar sequence of events can occur in such widely differing countries asJapan,Norway, Finland, Sweden, and Mexico suggest such bubbles and crashesare a general phenomenon.

How can this phenomenon be understood? The crucial issues we will focuson below are:

(i) What initiates a bubble?

(ii) What is the role of the banking system?

(iii) What causes a bubble to burst?

9.1.1 The risk-shifting problem

A simple example from Allen and Gale (2004) is developed to illustrate themodel in Allen and Gale (2000).1 They develop a theory based on rationalbehavior to try and provide some insight into these issues. Standard modelsof asset pricing assume people invest with their own money. We identify theprice of an asset in this benchmark case as the “fundamental.”A bubble is saidto occur when the price of an asset rises above this benchmark.2 If the peoplemaking investment decisions borrow money then because of default they areonly interested in the upper part of the distribution of returns of the riskyasset. As a result there is a risk-shifting problem and the price of the risky assetis bid up above the benchmark so there is a bubble.

1 For ease of exposition the example is slightly different from themodel presented in the paper.2 See Allen et al. (1993) for a discussion of the definition of fundamental and bubble.


In the example the people who make investment decisions do so with bor-rowed money. If they default there is limited liability. Lenders cannot observethe riskiness of the projects invested in so there is an agency problem. For thecase of real estate this representation of the agency problem is directly applic-able. For the case of stocks there are margin limits that prevent people directlyborrowing and investing in the asset. However, a more appropriate interpret-ation in this case is that it is institutional investors making the investmentdecisions. This group constitutes a large part of the market in many countries.The agency problem that occurs is similar to that with a debt contract. First,the people that supply the funds have little control over how they are invested.Second, the reward structure is similar to what happens with a debt contract. Ifthe assets the fundmanagers invest in do well, themanagers attract more fundsin the future and receive higher payments as a result. If the assets do badly thereis a limit to the penalty that is imposed on the managers. The worse that canhappen is that they are fired. This is analogous to limited liability (see Allenand Gorton 1993).

Initially there are two dates t = 1, 2. There are two assets in the example.The first is a safe asset in variable supply. For each 1 unit invested in this assetat date 1 the output is 1.5 at date 2. The second is a risky asset in fixed supplythat can be thought of as real estate or stocks. There is 1 unit of this risky asset.For each unit purchased at price P at date 1 the output is 6 with probability0.25 and 1 with probability 0.75 at date 2 so the expected payoff is 2.25. Thedetails of the two assets are given in Table 9.1.

The fundamental

Suppose each investor has wealth 1 initially and invests her ownwealth directly.Since everybody is risk neutral the marginal returns on the two assets must beequated:

2.25

PF= 1.5

1

Table 9.1. All agents in the model are assumed to be risk neutral.

Asset Supply Investment at date 1 Payoff at date 2

Safe Variable 1 1.5

Risky 1 P R ={

6 with prob. 0.25

1 with prob. 0.75ER = 2.25


or

PF = 2.25

1.5= 1.5.

The value of the asset is simply the discounted present value of the payoffwhere the discount rate is the opportunity cost of the investor. This is theclassic definition of the fundamental. The benchmark value of the asset is thus1.5 and any price above this is termed a bubble.

Intermediated case

Suppose next that investors have no wealth of their own. They can borrowto buy assets at a rate of 331

3 percent. The most they can borrow is 1. If theyborrow 1 they repay 1.33 if they are able to. If they are unable to pay thismuch the lender can claim whatever they have. As explained above lenderscan’t observe how loans are invested and this leads to an agency problem.

The first issue is can P = 1.5 be the equilibrium price?Consider what happens if an investor borrows 1 and invests in the safe asset.

Marginal return safe asset = 1.5 − 1.33

= 0.17.

Suppose instead that she borrows 1 and invests in the risky asset. She pur-chases 1/1.5 units. When the payoff is 6 she repays the principal and interestof 1.33 and keeps what remains. When it is 1 she defaults and the entire payoffgoes to the lender so she receives 0.

Marginal return risky asset = 0.25

(1

1.5× 6 − 1.33

)+ 0.75 × 0

= 0.25(4 − 1.33)

= 0.67.

The risky asset is clearly preferred when P = 1.5 since 0.67 > 0.17. Theexpected payoff of 1.5 on the investment in 1 unit of the safe asset is the sameas on the investment of 1/1.5 units of the risky asset. The risky asset is moreattractive to the borrower though. With the safe asset the borrower obtains0.17 and the lender obtains 1.33.With the risky asset the borrower obtains 0.67while the lender obtains 0.25×1.33+0.75×1× (1/1.5) = 1.5−0.67 = 0.83.The risk of default allows 0.5 in expected value to be shifted from the lender tothe borrower. This is the risk shifting problem. If the lender could prevent the


borrower from investing in the risky asset he would do so but he cannot sincethis is unobservable.

What is the equilibrium price of the risky asset given this agency problem?In an equilibrium where the safe asset is used, the price of the risky asset, P ,

will be bid up since it is in fixed supply, until the expected profit of borrowersis the same for both the risky and the safe asset:

0.25

(1

P× 6 − 1.33

)+ 0.75 × 0 = 1.5 − 1.33

so

P = 3.

There is a bubble with the price of the risky asset above the benchmark of 1.5.The idea that there is a risk-shifting problem when the lender is unable to

observe how the borrower invests the funds is not new (see, e.g. Jensen andMeckling 1976 and Stiglitz and Weiss 1981). However, it has not been widelyapplied in the asset pricing literature. Insteadof the standard result in corporatefinance textbooks that debt-financed firms are willing to accept negative netpresent value investments, the manifestation of the agency problem here isthat the debt-financed investors are willing to invest in assets priced abovetheir fundamental.

The amount of risk that is shifted depends on how risky the asset is. Thegreater the risk the greater the potential to shift risk and hence the higher theprice will be. To illustrate this consider the previous example but suppose thereturn on the risky asset is a mean-preserving spread of the original returns,as shown in Table 9.2. Now the price of the risky asset is given by

0.25

(1

P× 9 − 1.33

)+ 0.75 × 0 = 1.5 − 1.33

Table 9.2.

Asset Supply Investment at date 1 Payoff at date 2

Risky 1 P R ={

9 with prob. 0.25

0 with prob. 0.75ER = 2.25


so

P = 4.5.

More risk is shifted and as a result the price of the risky asset is bid up to aneven higher level.

It is interesting to note that in both the stock market boom of the 1920’s andthe one in the 1990’s the stocks that did best were “high-tech” stocks. In the1920’s it was radio stocks and utilities that were the star performers (see White1990). In the 1990’s it was telecommunications,media and entertainment, andtechnology stocks that did the best. It is precisely these stocks which have themost uncertain payoffs because of the nature of the business they are in.

One of the crucial issues is why the banks are willing to lend to the investorsgiven the chance of default. To see this consider again the casewhere the payoffson the risky asset are those in Table 9.1 and P = 3. In this case the quantityof the risky asset purchased when somebody borrows 1 is 1/P = 1/3. In theequilibria considered above the investors are indifferent between investing inthe safe and risky asset. Suppose for the sake of illustration the fixed supply ofthe risky asset is 1. The amount of funds depositors have is 10 and the numberof borrowers is 10. In the equilibrium where P = 3, 3 of the borrowers investin the risky asset and 7 in the safe in order for the fixed supply of 1 unit ofthe risky asset to be taken up. In this case 30 percent of borrowers are in riskyassets and 70 percent are in safe assets. A bank’s expected payoff from lendingone unit is then given by the following expression.

Bank’s expected payoff = 0.3[0.25 × 1.33 + 0.75 × (1/3) × 1] + 0.7[1.33]= 1.11.

The first term is the payoff to the bank from the 30 percent of investors inthe risky asset. If the payoff is 6, which occurs with probability 0.25, the loanand interest is repaid in full. If the payoff is 1, which occurs with probability0.75, the borrower defaults and the bank receives the entire proceeds from the1/3 unit owned by the borrower. The payoff is thus (1/3) × 1. The 70 percentof investors in the safe asset are able to pay off their loan and interest of 1.33in full.

If the banking sector is competitive the receipts from lending, 1.11, will bepaid out to depositors. In this case it is the depositors that bear the cost ofthe agency problem. In order for this allocation to be feasible markets must besegmented. The depositors and the banks must not have access to the assetsthat the investors who borrow invest in. Clearly if they did they would be betteroff to just invest in the safe asset rather than put their money in the bank.


9.1.2 Credit and interest rate determination

The quantity of credit and the interest rate have so far been taken as exogenous.These factors are incorporated in the example next to illustrate the relationshipbetween the amount of credit and the level of interest rates. We start with thesimplest casewhere the central bank determines the aggregate amount of creditB available to banks. It does this by setting reserve requirements and determin-ing the amount of assets available for use as reserves. For ease of exposition wedo not fully model this process and simply assume the central bank sets B. Thebanking sector is competitive. The number of banks is normalized at 1 andthe number of investors is also normalized to 1. Each investor will therefore beable to borrow B from each bank.

The return on the safe asset is determined by themarginal product of capitalin the economy. This in turn depends on the amount of the consumption goodx that is invested at date 1 in the economy’s productive technology to producef (x) units at date 2. The total amount that can be invested is B and the amountthat is invested at date 1 in the risky asset since there is 1 unit is P . Hence thedate 1 budget constraint implies that

x = B − P .It is assumed

f (x) = 3(B − P)0.5. (9.1)

Provided the market for loans is competitive the interest rate r on bank loanswill be such that

r = f ′(B − P) = 1.5(B − P)−0.5. (9.2)

At this level borrowing and investing in the safe asset will not yield any profitsfor investors. If r was lower than this there would be an infinite demand forbank loans to buy the safe asset. If r was higher than this there would bezero demand for loans and nobody would invest in the safe asset but this is acontradiction since f ′(0) = ∞.

The amount the investors will be prepared to pay for the risky asset assumingits payoffs are as in Table 9.1 is then given by

0.25

(1

P× 6 − r

)+ 0.75 × 0 = 0.

Using (9.2) in this,

P = 4(B − P)0.5.


Solving for P gives

P = 8(−1 + √1 + 0.25B). (9.3)

When B = 5 then P = 4 and r = 1.5. The relationship between P andB is shown by the solid line in Figure 9.1. By controlling the amount ofcredit the central bank controls the level of interest rates and the level ofasset prices. Note that this relationship is different from that in the standardasset pricing model when the price of the risky asset is the discounted expectedpayoff.

PF = 2.25

r.

This case is illustrated by the dotted line in Figure 9.1. A comparison of thetwo cases shows that the fundamental is relatively insensitive to the amountof credit compared to the case where there is an agency problem. Changes inaggregate credit can cause relatively large changes in asset prices when there isan agency problem.

0

1

2

3

4

5

6

7

00.

40.

81.

21.

6 22.

42.

83.

23.

6 44.

44.

85.

25.

6 66.

46.

87.

27.

6 8

B

P

AgencyFundamental

Figure 9.1. Credit and asset prices (from Figure 1 of Allen and Gale 2004).


9.1.3 Financial risk

The previous section assumed that the central bank could determine theamount of credit B. In practice the central bank has limited ability to con-trol the amount of credit and this means B is random. In addition there maybe changes of policy preferences, changes of administration, and changes inthe external environment which create further uncertainty about the level ofB. This uncertainty is particularly great in countries undergoing financial lib-eralization. In order to investigate the effect of this uncertainty an extra periodis added to the model. Between dates 1 and 2 everything is the same as before.Between dates 0 and 1 the only uncertainty that is resolved is about the levelof B at date 1. Thus between dates 0 and 1 there is financial uncertainty. Theuncertainty about aggregate credit B at date 1 causes uncertainty about pricesat date 1. Given that investors are borrowing from banks at date 0 in the sameway as before this price uncertainty again leads to an agency problem and riskshifting. The price of the risky asset at date 0 will reflect this price uncertaintyand can lead the asset price to be even higher than at date 1.

Suppose that there is a 0.5 probability that B = 5 and a 0.5 probability thatB = 7 at date 1. Then using (9.2) and (9.3) the prices and interest rates are asshown in Table 9.3.

Table 9.3.

Probability B P r

0.5 5 4 1.50.5 7 5.27 1.14

The pricing equation at date 0 is

0.5

(1

P0× 5.27 − r0

)+ 0.5 × 0 = 0,

where r0, the date 0 interest rate, is given by (9.2) with B and P replaced by B0

and P0. Substituting for r0 and simplifying

P0 = 5.27

1.5(B0 − P0)0.5.

Taking B0 = 6 and solving for r0 and P0 gives

r0 = 1.19

P0 = 4.42.


As when the uncertainty is due to variations in asset returns, the greater thefinancial uncertainty the greater is P0. Consider a mean preserving spread onthe financial uncertainty so that Table 9.3 is replaced by Table 9.4.

Table 9.4.

Probability B P r

0.5 4 3.14 1.810.5 8 5.86 1.03

In this case it can be shown

r0 = 1.27

P0 = 4.61.

The risk-shifting effect operates for financial risk in the same way as it does forreal risk. Although the expected payoff at date 2 is only 2.25 the price of therisky asset at date 1 in the last case is 4.61. The possibility of credit expansionover a period of years may create a great deal of uncertainty about how highthe bubble may go and when it may collapse. This is particularly true wheneconomies are undergoing financial liberalization. As more periods are addedit is possible for the bubble to become very large. The market price can bemuch greater than the fundamental.

9.1.4 Financial fragility

The examples in the previous section illustrated that what is important indetermining the risky asset’s price at date 0 is expectations about aggregatecredit at date 1. If aggregate credit goes up then asset prices will be high anddefault will be avoided.However, if aggregate credit goes down then asset priceswill be low and default will occur. The issue here is what is the dynamic path ofaggregate credit. The point is that the expectation of credit expansion is alreadytaken into account in the investors’ decisions about how much to borrow andhow much to pay for the risky asset. If credit expansion is less than expected,or perhaps simply falls short of the highest anticipated levels, the investorsmaynot be able to repay their loans and default occurs. In Allen and Gale (2000) itis shown that even if credit is always expanded then there may still be default.In fact it is shown that there are situations where the amount of credit willbe arbitrarily close to the upper bound of what is anticipated and widespreaddefault is almost inevitable.

9.2 Banking Crises and Negative Bubbles 247

9.2 BANKING CRISES AND NEGATIVE BUBBLES

In the previous section we focused on how asset prices could get too highbecause of an agency problem between lenders and the people making invest-ment decisions. In this section we consider what happens when asset prices arelow relative to the fundamental. An important feature of many of the historicand recent banking crises is the collapse in asset prices that accompanies them.The purpose of this section is to consider this phenomenon using the model ofAllen and Gale (1998, 2004).We start by developing a simple model and derivethe optimal allocation of resources. If there is a market for risky assets thatallows banks to sell their assets then the allocation is not efficient. The simul-taneous liquidation of all banks’ assets that accompanies a crisis leads to a neg-ative bubble and inefficient risk sharing. However, by adopting an appropriatemonetary policy a central bank can implement the optimal allocation.

9.2.1 The model

Time is divided into three periods t = 0, 1, 2. There are two types of assets,a safe asset and a risky asset, and a consumption good. The safe asset can bethought of as a storage technology,which transforms one unit of the consump-tion good at date t into one unit of the consumption good at date t + 1. Therisky asset is represented by a stochastic production technology that trans-forms one unit of the consumption good at date t = 0 into R units of theconsumption good at date t = 2, where R is a non-negative random variablewith

R ={RH with probability π

RL with probability 1 − π .

At date 1 depositors observe a signal, which can be thought of as a leading eco-nomic indicator, similarly to Gorton (1988). This signal predicts with perfectaccuracy the value of R that will be realized at date 2. Initially it is assumed thatconsumption can be made contingent on the leading economic indicator, andhence onR. Subsequently,we consider what happens when banks are restrictedto offering depositors a standard deposit contract, that is, a contract which isnot explicitly contingent on the leading economic indicator.

There is a continuum of ex ante identical depositors (consumers) who havean endowment of 1 of the consumption good at the first date and none at thesecond and third dates. Consumers are uncertain about their time preferences.Some will be early consumers, who only want to consume at date 1, and some


will be late consumers,who onlywant to consume at date 2.At date 0 consumersknow the probability of being an early or late consumer, but they do not knowwhich group they belong to. All uncertainty is resolved at date 1 when eachconsumer learns whether he is an early or late consumer and what the returnon the risky asset is going to be. For simplicity, we assume that there are equalnumbers of early and late consumers and that each consumer has an equalchance of belonging to each group. Then a typical consumer’s expected utilitycan be written as

λU (c1) + (1 − λ)U (c2) (9.4)

where ct denotes consumption at date t = 1, 2. The period utility func-tionsU (·) are assumed to be twice continuously differentiable, increasing andstrictly concave. A consumer’s type is not observable, so late consumers canalways imitate early consumers. Therefore, contracts explicitly contingent onthis characteristic are not feasible.

The role of banks is tomake investments on behalf of consumers.We assumethat only banks can hold the risky asset. This gives the bank an advantage overconsumers in two respects. First, the banks can hold a portfolio consisting ofboth types of assets, which will typically be preferred to a portfolio consistingof the safe asset alone. Second, by pooling the assets of a large number ofconsumers, the bank can offer insurance to consumers against their uncertainliquidity demands, giving the early consumers some of the benefits of thehigh-yielding risky asset without subjecting them to the volatility of the assetmarket.

Free entry into the banking industry forces banks to compete by offeringdeposit contracts that maximize the expected utility of the consumers. Thus,the behavior of the banking industry can be represented by an optimal risk-sharing problem. A variety of different risk-sharing problems can be usedto represent different assumptions about the informational and regulatoryenvironment.

9.2.2 Optimal risk sharing

Initially consider the casewhere banks canwrite contracts inwhich the amountthat can be withdrawn at each date is contingent on R. This provides a bench-mark for optimal risk sharing. Since the risky asset return is not known untilthe second date, the portfolio choice is independent of R, but the paymentsto early and late consumers, which occur after R is revealed, will depend onit. Let y and x = 1 − y denote the representative bank’s holding of the risky


and safe assets, respectively. The deposit contract can be represented by a pairof functions, c1(R) and c2(R) which give the consumption of early and lateconsumers conditional on the return to the risky asset.

The optimal risk-sharing problem can be written as follows.

max E[λU (c1(R)) + (1 − λ)U (c2(R))]s.t. (i) y + x ≤ 1;

(ii) λc1(R) ≤ y ;(iii) λc1(R) + (1 − λ)c2(R) ≤ y + Rx ;(iv) c1(R) ≤ c2(R).

(9.5)

The first constraint says that the total amount invested must be less than orequal to the amount deposited. There is no loss of generality in assuming thatconsumers deposit their entire wealth with the bank, since anything they cando the bank can do for them. The second constraint says that the holding ofthe safe asset must be sufficient to provide for the consumption of the earlyconsumers at date 1. The bankmaywant to hold strictlymore than this amountand roll it over to the final period, in order to reduce the uncertainty of the lateconsumers. The next constraint, together with the preceding one, says that theconsumption of the late consumers cannot exceed the total value of the riskyasset plus the amount of the safe asset left over after the early consumers arepaid off, that is,

(1 − λ)c2(R) ≤ (y − λc1(R)) + Rx . (9.6)

The final constraint is the incentive compatibility constraint. It says that forevery value of R, the late consumers must be at least as well off as the earlyconsumers. Since late consumers are paid off at date 2, an early consumercannot imitate a late consumer. However, a late consumer can imitate an earlyconsumer, obtain c1(R) at date 1, and use the storage technology to providehimself with c1(R) units of consumption at date 2. It will be optimal to do thisunless c1(R) ≤ c2(R) for every value of R.

The following assumptions aremaintained throughout the section to ensureinterior optima. The preferences and technology are assumed to satisfy theinequalities

E[R] > 1 (9.7)

and

U ′(0) > E[U ′(RE)R]. (9.8)


The first inequality ensures a positive amount of the risky asset is held whilethe second ensures a positive amount of the safe asset is held.

An examination of the optimal risk-sharing problem shows us that theincentive constraint (iv) can be dispensed with. To see this, suppose that wesolve the problem subject to the first three constraints only. A necessary condi-tion for an optimum is that the consumption of the two types be equal, unlessthe feasibility constraint λc1(R) ≤ y is binding, in which case it follows fromthe first-order conditions that c1(R) ≤ c2(R). Thus, the incentive constraintwill always be satisfied if we optimize subject to the first three constraints onlyand the solution to (9.5) is the first-best allocation.

It can be shown that the solution to the problem is

c1(R) = c2(R) = y + Rx ify

λ≥ Rx

1 − λ, (9.9)

c1(R) = y/λ, c2(R) = Rx/(1 − λ) ify

λ<Rx

1 − λ, (9.10)

y + x = 1 (9.11)

E[U ′(c1(R))] = E[U ′(c2(R))R]. (9.12)

(See Allen and Gale 1998 for a formal derivation of this.)The optimal allocation is illustrated in Figure 9.2. When the signal at date 1

indicates that R = 0 at date 2, both the early and late consumers receive y sincey is all that is available and it is efficient to equate consumption given the formof the objective function. The early consumers consume their shareλy at date 1with the remaining (1−λ)y carried over until date 2 for the late consumers. AsR increases both groups can consume more until y/λ = Rx/(1−λ). ProvidedR < (1 − λ)y/λx ≡ R the optimal allocation involves carrying over some of

c2(R)

c1(R)d = y/λ

R = (1–λ)y/λx

y

ct(R)

R

Figure 9.2. Optimal risk sharing.


the liquid asset to date 2 to supplement the low returns on the risky asset forlate consumers. When the signal indicates that R will be high at date 2 (i.e.R ≥ (1 − λ)y/λx ≡ R), then early consumers should consume as much aspossible at date 1 which is y/λ since consumption at date 2 will be high in anycase. Ideally, the high date 2 output would be shared with the early consumersat date 1, but this is not technologically feasible. It is only possible to carryforward consumption, not bring it back from the future.

To illustrate the operation of the optimal contract, we adopt the followingnumerical example:

U = ln(ct );

EU = 0.5 ln(c1) + 0.5 ln(c2); (9.13)

R ={

2 with probability 0.9;0.6 with probability 0.1.

For these parameters, it can readily be shown that (y , x) = (0.514, 0.486) andR = 1.058. The levels of consumption are

c1(2) = 1.028; c2(2) = 1.944 with probability 0.9;c1(0.6) = c2(0.6) = 0.806 with probability 0.1.

The level of expected utility achieved is EU = 0.290.

9.2.3 Optimal deposit contracts

Suppose next that contracts can’t be explicitly conditioned on R. Let d denotethe fixed payment promised to the early consumers at date 1. Initially weassume there is no market for the long term asset. If the bank is unable tomake the payment d to all those requesting it at date 1 then the short-termasset that it does have is split up equally between them. Since the bankingsector is competitive and the objective of the bank is to maximize the expectedutility of depositors the late consumers will always be paid whatever is availableat the last date. In that case, in equilibrium the early and late consumers willhave the same consumption.

It can straightforwardly be seen in the example that the optimal allocationcan be implemented using a deposit contract. To do this the bank chooses(y , x) = (0.514, 0.486) and sets d = 1.028. Anything left over at date 2 isdistributed equally among the remaining depositors. When R = RH = 2the bank uses all the short asset to pay out to its early consumers and they


receive c1(2) = d = 0.514/0.5 = 1.028. The late consumers receive c2(2) =0.486 × 2/0.5 = 1.944.

When R = RL = 0.6 the late consumers can calculate that if all the earlyconsumers were to receive d = 1.028 and exhaust all the short asset then theamount remaining for each late consumer at date 2 would be

0.6 × 0.486

0.5= 0.583 < 1.028.

Thus some of the late consumers will also withdraw at date 1. This meansthe bank will not be able to satisfy all those withdrawing. Since there is nomarket for the long asset, what happens is that the available proceeds are splitequally among those asking to withdraw, as explained above. Suppose α(0.6)late consumers withdraw early, then the λ + α(0.6) withdrawing early and the1 − λ − α(0.6) withdrawing late will have the same expected utility when

y

λ + α(0.6)= Rx

1 − λ − α(0.6).

Substituting

0.514

0.5 + α(0.6)= 0.6 × 0.486

1 − 0.5 − α(0.6)

and solving for α(0.6) gives

α(0.6) = 0.138.

Hence the consumption of everybody is

c1(0.6) = 0.514

0.5 + 0.138= c2(0.6) = 0.6 × 0.486

1 − 0.5 − 0.138= 0.806.

In Allen and Gale (1998) it is shown more generally that the optimalallocation can be implemented using a deposit contract.

9.2.4 An asset market

Suppose next that there is a competitive market for liquidating the long-termasset for price P . If the bank can make the payment d to the depositors whorequest to withdraw at date 1 then it continues until date 2. But if the bankis unable to do this then it goes bankrupt and its assets are liquidated anddistributed on a pro rata basis among its depositors. Then the standard deposit


contract promises the early consumers either d or, if that is infeasible, an equalshare of the liquidated assets.

The participants in the long-term asset market are the banks, who use it toobtain liquidity, and a large number of wealthy, risk neutral speculators whohope to make a profit in case some bank has to sell off assets cheaply to getliquidity. The speculators hold some cash (the safe asset) in order to purchasethe risky asset when its price at date 1 is sufficiently low. The return on thecash is low, but it is offset by the prospect of speculative profits when the priceof the risky asset falls below its fundamental value. Suppose the risk neutralspeculators hold some portfolio (ys , xs). They cannot short sell or borrow. Inequilibrium they will be indifferent between the portfolio (ys , xs) and puttingall their money in the risky asset.

The impact of introducing the asset market can be illustrated usingFigure 9.3. The graphs in this figure represent the equilibrium consump-tion levels of early and late consumers, respectively, as a function of the riskyasset return R. For high values of R (i.e. R ≥ R∗), there is no possibility ofa bank run. The consumption of early consumers is fixed by the standarddeposit contract at c1(R) = d and the consumption of late consumers isgiven by the budget constraint c2(R) = (y + Rx − d)/(1 − λ). For lowervalues of R (R < R∗), it is impossible to pay the early consumers the fixedamount d promised by the standard deposit contract without violating thelate consumers’ incentive constraint

y + Rx − λd

1 − λ≥ d

and a bank run inevitably ensues. The terms of the standard deposit contractrequire the bank to liquidate all of its assets at the second date if it cannot pay

c2(R )

c1(R )

R*

y

y+ys

d

R0

ct(R )

R

Figure 9.3. Consumption without Central Bank intervention.


d to every depositor who demands it. Since late withdrawers always receiveas much as the early consumers by incentive compatibility, the bank has toliquidate all its assets unless it can give at least d to all consumers. The valueof R∗ is determined by the condition that the bank can just afford to giveeveryone d so

(1 − λ)d = y + R∗x − λd

or

R∗ = d − yx

.

BelowR∗ it is impossible for the bank to pay all the depositors d , and the onlyalternative is to liquidate all its assets at the first date and pay all consumers lessthan d . Since a late withdrawer will receive nothing, all consumers will chooseto withdraw their deposits at the second date.

There is a discontinuity in the consumption profiles at the critical value ofR∗ that marks the upper bound of the interval in which runs occur. The reasonfor this discontinuity is the effect of asset sales on the price of the risky asset.By selling the asset, the bank drives down the price, thus handing a windfallprofit to the speculators and a windfall loss to the depositors. This windfallloss is experienced as a discontinuous drop in consumption.

The pricing of the risky asset at date 1 is shown in Figure 9.4. For R > R∗the speculators continue to hold both assets and are indifferent between them.Since one unit of the safe asset is worth 1 in the last period, the fundamentalvalue of each unit of the risky asset is R/1 = R. For R < R∗ the banks areforced to liquidate all of their assets. Now the speculators can use their cash to

P(R)=R

R0 R*

P(R)

R

yS /x

Figure 9.4. Asset pricing without Central Bank intervention.


buy the risky asset. Provided R is such that R0 < R < R∗ where

R0 = ySx

,

the speculators will want to use all of their cash to buy the risky asset. Theamount of cash in the market yS is insufficient to pay the fundamental value ofthe risky asset, so the price is determined by the ratio of the speculators’ cashto the bank’s holding of the risky asset

P(R) = ySx

.

For R0 < R < R∗ there is “cash-in-the-market pricing” and the price of therisky asset is below its fundamental value. In other words there is a negativebubble. For small values of R (R < R0) the fundamental value of the riskyasset is less than the amount of cash in the market, so the asset price is equalto the fundamental value once again.

Since the price is independent of R for R0 < R < R∗ consumption isindependent of R in this interval as Figure 9.3 indicates. The consumptionavailable at date 1 consists of the bank’s holding of the safe asset, y , and thespeculators’ holding ys . This is split among the early and late consumers soeach receives y + ys .

To sum up, introducing a market for the risky asset has a number of import-ant implications. It allows the bank to liquidate all its assets to meet thedemands of the early withdrawers, but this has the effect of making the situ-ation worse. First, because a bank run exhausts the bank’s assets at date 1, alate consumer who waits until date 2 to withdraw will be left with nothing, sowhenever there is a bank run, it will involve all the late consumers and not justsome of them. Second, if the market for the risky asset is illiquid, the sale ofthe representative bank’s holding of the risky asset will drive down the price,thus making it harder to meet the depositors’ demands.

The all-or-nothing character of bank runs is, of course, familiar from thework of Diamond and Dybvig (1983). The difference is that in the presentmodel bank runs are not “sunspot” phenomena: they occur only when there isno other equilibrium outcome possible. Furthermore, the deadweight cost of abank run in this case is endogenous. There is a cost resulting from suboptimalrisk sharing.When the representative bank is forced to liquidate the risky asset,it sells the asset at a low price. This is a transfer of value to the purchasers ofthe risky asset, not an economic cost. The deadweight loss arises because thetransfer occurs in bad states when the consumers’ consumption is already low.In other words, the market is providing negative insurance.


The outcome with an asset market is in fact Pareto worse than the optimalallocation. The bank depositors are clearly worse off since they have lowerconsumption for R0 ≤ R ≤ R∗ and the speculators are indifferent. Thiscan be illustrated using a variant of the numerical example above. Supposethat the wealth of the speculators Ws = 1 and that the other parameters areas before. The optimal contract for depositors has (y , x) = (0.545, 0.455),R0 = 0.070,R∗ = 1.193, with P(R) = 0.070 for R0 < R < R∗ andEU = 0.253. For the speculators (ys , xs) = (0.032, 0.968) and their expectedutility is EUs = 1.86. They receive the same level of expected utility as theywould if they invested all their funds in the risky asset. Note that the depos-itors are significantly worse off in this equilibrium compared to the allocationcorresponding to the solution to (9.5) where EU = 0.290.

9.2.5 Optimal monetary policy

The inefficiency in the allocation when there is an asset market arises fromthe negative bubble in asset prices. A central bank can prevent the collapse inasset prices and ensure that the allocation is the same as in Figure 9.2 by anappropriate intervention. The essential idea behind the policy that implementsthe solution to (9.5) is that the central bank enters into a repurchase agreement(or a collateralized loan) with the representative bank, whereby the bank sellssome of its assets to the central bank at date 1 in exchange for money andbuys them back for the same price at date 2. By providing liquidity in this way,the central bank ensures that the representative bank does not suffer a loss byliquidating its holdings of the risky asset prematurely.

We assume that the standard deposit contract is now written in nominalterms. The contract promises depositors a fixed amount of money D in themiddle period and pays out the remaining value of the assets in the last period.The price level at date t in state R is denoted by pt (R) and the nominal priceof the risky asset at date 1 in state R is denoted by P(R). We want the riskyasset to sell for its fundamental value, so we assume that P(R) = p1(R)R. Atthis price, the safe and risky assets are perfect substitutes. Let (y , x) be theportfolio corresponding to the solution of (9.5) and let (c1(R), c2(R)) be thecorresponding consumption allocations. For large values of R,we have c1(R) =y/λ < c2(R) = Rx/(1−λ); for smaller valueswehave c1(R) = c2(R) = y+Rx .Implementing this allocation requires introducing contingencies throughpricevariation: p1(R)c1(R) = D < p2(R)c2(R) for R > R and p1(R)c1(R) = D =p2(R)c2(R) for R < R. These equations determine the values of p1(R) andp2(R) uniquely. It remains only to determine the value of sales of assets andthe size of the bank run.


In the event of a bank run, only the late consumers who withdraw earlywill end up holding cash, since the early consumers want to consume theirentire liquidated wealth immediately. If α(R) is the fraction of late consumerswho withdraw early, then the amount of cash injected into the system mustbe α(R)D. For simplicity, we assume that the amount of cash injected is aconstantM and this determines the “size” of the run α(R). Since the safe assetand the risky asset are perfect substitutes at this point, it does not matter whichassets the representative bank sells to the central bank as long as the nominalvalue equals M . The representative bank enters into a repurchase agreementunder which it sells assets at date 1 for an amount of cash equal to M andrepurchases them at date 2 for the same cash value.

At the prescribed prices, speculators will not want to hold any of the safeassets, so ys = 0 and xs = Ws .

It is easy to check that all the equilibrium conditions are satisfied: depositorsand speculators are behaving optimally at the given prices and the feasibilityconditions are satisfied.

To summarize, the central bank can implement the solution to (9.5) byentering into a repurchase agreement with the representative bank at date1. Given the allocation {(y , x), c1(R), c2(R)}, corresponding to the solution of(9.5), the equilibriumvalues of prices are given by the conditions p1(R)c1(R) =D < p2(R)c2(R) for R > R, and p1(R)c1(R) = D = p2(R)c2(R) for R < R.There is a fixed amount of money M injected into the economy in the eventof a run and the fraction of late withdrawers who “run” satisfies α(R)D = M .The price of the risky asset at date 1 satisfies p1(R)R = P(R) and the optimalportfolio of the speculators is (ys , xs) = (0,Ws).

It can be seen that the central bank intervention ensures that the risky asset’sprice is always equal to its fundamental value. This means that speculatorsdo not profit and depositors do not lose for R0 ≤ R ≤ R∗. As a result it isstraightforward to show that the allocation is (strictly) Pareto-preferred to theequilibrium of the model with asset markets.

This can be illustrated with the numerical example. Recall that the solutionto (9.5) has (y , x) = (0.514, 0.486), R = 1.058 and EU = 0.290. SupposeD = 1.028. For R ≥ R = 1.058 then p1(R) = p2(R) = 1. For R < R = 1.058the price levels at the two dates depend on the level of R. For the state RL =0.6, c1(0.6) = c2(0.6) = 0.806 so p1(0.6) = p2(0.6) = 1.028/0.806 = 1.275.The lower the value of R, the higher pt (R), so that consumption is lowered byraising the price level. Also P(R) = 1.028 × 0.6 = 0.617. The fraction of lateconsumers who withdraw from the bank and hold money will be determinedbyM . SupposeM = 0.1, then α(R) = 0.1/1.028 = 0.097. For the speculators(ys , xs) = (0, 1) and their expected utility is EUs = 1.86. The equilibrium with


central bank intervention is clearly Pareto-preferred to the market equilibriumwithout intervention.


This chapter has argued that monetary policy can have an effect on asset pricesin two important ways. The first is that when there is an agency problembetween banks and the people they lend to who make investment decisionsasset prices can rise above their fundamental. The agency problem means thatinvestors choose riskier projects than they otherwise would and bid up prices.The greater the risk the larger this bubble can become. It is not only the riskthat is associated with real asset returns that can cause a bubble but also thefinancial risk associatedwith the uncertainties of monetary policy and particu-larly financial liberalization. The first important conclusion is that the centralbank should keep such uncertainties to a minimum. The less uncertainty, theless the magnitude of the positive bubble.

The second problem occurs when asset prices fall. If this fall causes banksto liquidate assets simultaneously then asset prices can fall below their funda-mental value. In other words there is a negative bubble. This bubble can alsobe very damaging. In this case it is desirable for the central bank to step inand provide liquidity and prevent asset prices falling below their fundamentalvalue. They can do this by lending against the banks’ assets.

The central bank has a complicated task to prevent both types of bubble.Moreover it is important for it to correctly identify which is the relevant prob-lem and the appropriate policy to solve it otherwise the situation will only beexacerbated.

REFERENCES

Allen, F. and D. Gale (1998). “Optimal Financial Crises,” Journal of Finance 53, 1245–1284.

Allen, F. and D. Gale (2000). “Bubbles and Crises,” Economic Journal 110, 236–255.Allen, F. and D. Gale (2004). “Asset Price Bubbles and Monetary Policy,” in M. Desai

and Y. Said (eds.), Global Governance and Financial Crises, New York and London:Routledge, Chapter 3, 19–42.

Allen, F. and G. Gorton (1993). “Churning Bubbles,” Review of Economic Studies 60,813–836.

Allen, F., S. Morris, and A. Postlewaite (1993). “Finite Bubbles with Short SaleConstraints andAsymmetric Information,” Journal of Economic Theory 61, 206–229.

References 259


Drees, B. andC. Pazarbasioglu (1995).“TheNordic BankingCrises: Pitfalls in FinancialLiberalization?” Working Paper 95/61, International Monetary Fund, Washington,DC.

Englund, P. and V. Vihriälä (2006). “Financial Crises in Developed Economies: TheCases of Finland and Sweden,” Chapter 3 in L. Jonung (ed.), Crises, MacroeconomicPerformance and Economic Policies in Finland and Sweden in the 1990s: AComparativeApproach, forthcoming.

Frankel, J. (1993). “The Japanese Financial System and the Cost of Capital,” in S. Takagi(ed.), Japanese Capital Markets: New Developments in Regulations and Institutions,Oxford: Blackwell, 21–77.


Heiskanen,R. (1993).“TheBankingCrisis in theNordicCountries.”Kansallis EconomicReview 2, 13–19.

Jensen,M. andW. Meckling (1976). “Theory of the Firm:Managerial Behavior,AgencyCost and Ownership Structure,” Journal of Financial Economics 3, 305–360.

Kindleberger, C. (1978). Manias, Panics, and Crashes: A History of Financial Crises,New York, NY: Basic Books.

Mishkin, F. (1997). “Understanding Financial Crises: A Developing Country Per-spective.” Annual World Bank Conference on Development Economics 1996, 29–61,Washington, DC: The International Bank for Reconstruction and Development.

Stiglitz, J. and A. Weiss (1981). “Credit Rationing in Markets with ImperfectInformation,”American Economic Review 71, 393–410.

Tschoegl, A. (1993). “Modeling the Behaviour of Japanese Stock Indices,” in S. Takagi(ed.), Japanese Capital Markets: New Developments in Regulations and Institutions,Oxford: Blackwell, 371–400.

White, E. (1990). Crashes and Panics: The Lessons from History, Homewood, IL:Dow Jones Irwin.

10

Contagion

Financial contagion refers to the process by which a crisis that begins in oneregion or country spreads to an economically linked region or country. Some-times the basis for contagion is provided by information. Kodres and Pritsker(2002), Calvo and Mendoza (2000a, b) and Calvo (2002) show how asymmet-ric information can give rise to contagion between countries that are affectedby common fundamentals. An example is provided by asset markets in twodifferent countries. A change in prices may result from a common shock thataffects the value of assets in both countries or it may result from an idiosyn-cratic shock that either has no effect on asset values (a liquidity shock) or thataffects only one country. Because the idiosyncratic shock can be mistaken forthe common shock, a fall in prices in one country may lead to a self-fulfillingexpectation that prices will fall in the other country. In that case, an unneces-sary and possibly costly instability arises in the second country because of anunrelated crisis in the first.

A second type of contagion is explored in this chapter. The possibility of thiskind of contagion arises from the overlapping claims that different regions orsectors of the banking system have on one another. When one region suffersa banking crisis, the other regions suffer a loss because their claims on thetroubled region fall in value. If this spillover effect is strong enough, it cancause a crisis in the adjacent regions. In extreme cases, the crisis passes fromregion to region, eventually having an impact on a much larger area than theregion in which the initial crisis occurred.

The central aim of this chapter is to provide some microeconomic founda-tions for financial contagion. The model developed below is not intended tobe a description of any particular episode. It has some relevance to the recentAsian financial crisis. For example,VanRijckeghem andWeder (2000) considerthe interlinkages between banks in Japan and emerging countries inAsia, LatinAmerica, and Eastern Europe. As one might expect, the Japanese banks had themost exposure in Asian emerging economies. When the Asian crisis started,in Thailand in July 1997, the Japanese banks withdrew funds not only fromThailand but also from other emerging countries, particularly from countriesin Asia, where they had the most exposure. In this way, the shock of the crisis

Chapter 10. Contagion 261

in Thailand spread to other Asian countries. European and North Americanbanks also reacted to the Asian crisis by withdrawing funds fromAsia, but theyactually increased lending to Latin America and Eastern Europe. Ultimately, itwas Asian countries that were most affected by the initial shock.

The model we describe below has the closest resemblance to the bankingcrises in the United States in the late nineteenth and early twentieth centuries(Hicks 1989). As we saw in Chapter 1, banks in the Midwest and other regionsof the US held deposits in New York banks. These linkages provide a channelfor spillovers between the banks if there is a financial crisis in one region.

In order to focus on the role of one particular channel for financial con-tagion, in what follows we exclude other propagation mechanisms that maybe important for a fuller understanding of financial contagion. In particular,we assume that agents have complete information about their environment.As was mentioned above, incomplete information may create another channelfor contagion. We also exclude the effect of international currency markets inthe propagation of financial crises from one country to another. The role ofcontagion in currency crises has been extensively studied and is summarizedin an excellent survey by Masson (1999).

We use our standard model with a number of slight variations that allow usto focus on contagion through interlinkages. In particular, we assume that thelong asset is liquidated using a liquidation technology rather than being soldat the market price. There are three dates t = 0, 1, 2 and a large number ofidentical consumers, each of whom is endowed with one unit of a homoge-neous good that can be consumed or invested. At date 1, the consumers learnwhether they are early consumers, who only value consumption at date 1, orlate consumers,who only value consumption at date 2. Uncertainty about theirpreferences creates a demand for liquidity.

Banks have a comparative advantage in providing liquidity. At the first date,consumers deposit their endowments in the banks, which invest them onbehalf of the depositors. In exchange, depositors are promised a fixed amountof consumption at each subsequent date, depending on when they choose towithdraw. The bank can invest in two assets. There is a short-term asset thatpays a return of one unit after one period and there is a long-term asset thatcan be liquidated for a return r < 1 after one period or held for a return ofR > 1 after two periods. The long asset has a higher return if held to maturity,but liquidating it in the middle period is costly, so it is not very useful forproviding consumption to early consumers. The banking sector is perfectlycompetitive, so banks offer risk-sharing contracts that maximize depositors’ex ante expected utility, subject to a zero-profit constraint.

Using this framework, we construct a simple model in which small shockslead to large effects by means of contagion. More precisely, a shock within

262 Chapter 10. Contagion

a single sector has effects that spread to other sectors and lead eventually to aneconomy-wide financial crisis. This form of contagion is driven by real shocksand real linkages between regions. As we have seen, one view is that financialcrises are purely random events, unrelated to changes in the real economy(Kindleberger 1978). The modern version of this view, developed by Diamondand Dybvig (1983) and others, is that bank runs are self-fulfilling prophecies.The disadvantage of treating contagion as a “sunspot” phenomenon is that,without some real connection between different regions, any pattern of correl-ations is possible. So sunspot theories do not provide a causal link betweencrises in different regions. We adopt the alternative view that financial crisesare an integral part of the business cycle (Mitchell 1941; Gorton 1988; Allenand Gale 1998) and show that, under certain circumstances, any equilibriumof the modelmust be characterized by contagion.

The economy consists of a number of regions. The number of early and lateconsumers in each region fluctuates randomly, but the aggregate demand forliquidity is constant. This allows for interregional insurance as regions withliquidity surpluses provide liquidity for regions with liquidity shortages. Oneway to organize the provisionof insurance is through the exchange of interbankdeposits. Suppose that region A has a large number of early consumers whenregion B has a low number of early consumers, and vice versa. Since regions Aand B are otherwise identical, their deposits are perfect substitutes. The banksexchange deposits at the first date, before they observe the liquidity shocks.If region A has a higher than average number of early consumers at date 1,then banks in region A can meet their obligations by liquidating some of theirdeposits in the banks of region B. Region B is happy to oblige, because it hasan excess supply of liquidity, in the form of the short asset. At the final date,the process is reversed, as banks in region B liquidate the deposits they hold inregion A to meet the above-average demand from late consumers in region B.

Inter-regional cross holdings of deposits workwell as long as there is enoughliquidity in the banking system as a whole. If there is an excess demand forliquidity,however, thefinancial linkages causedby these cross holdings can turnout to be a disaster. While cross holdings of deposits are useful for reallocatingliquidity within the banking system, they cannot increase the total amountof liquidity. If the economy-wide demand from consumers is greater thanthe stock of the short asset, the only way to provide more consumption is toliquidate the long asset. There is a limit to howmuch can be liquidated withoutprovoking a run on the bank, however, so if the initial shock requires morethan this buffer, there will be a run on the bank and the bank is forced intobankruptcy. Banks holding deposits in the defaulting bank will suffer a capitalloss, which may make it impossible for them to meet their commitments toprovide liquidity in their region. Thus, what began as a financial crisis in one

10.1 Liquidity Preference 263

region will spread by contagion to other regions because of the cross holdingsof deposits.

Whether the financial crisis does spread depends crucially on the pattern ofinter-connectedness generated by the cross holdings of deposits. We say thatthe interbank network is complete if each region is connected to all the otherregions and incomplete if each region is connected with a small number ofother regions. In a complete network, the amount of interbank deposits thatany bank holds is spread evenly over a large number of banks. As a result,the initial impact of a financial crisis in one region may be attenuated. In anincomplete network, on the other hand, the initial impact of the financial crisisis concentrated in the small number of neighboring regions, with the resultthat they easily succumb to the crisis too. As each region is affected by thecrisis, it prompts premature liquidation of long assets, with a consequent lossof value, so that previously unaffected regions find that they too are affected.

It is important to note the role of a free rider problem in explaining theprocess of contagion. Cross holdings of deposits are useful for redistributingliquidity, but they do not create liquidity. So when there is excess demand forliquidity in the economy as a whole each bank tries to meet external demandsfor liquidity by drawing down its deposits in another bank. In other words,each bank is trying to “pass the buck” to another bank. The result is that all theinterbank deposits disappear and no one gets any additional liquidity.

The only solution to a global shortage of liquidity (withdrawals exceed shortassets), is to liquidate long assets.Aswehave seen,eachbankhas a limited bufferthat it can access by liquidating the long asset. If this buffer is exceeded, the bankmust fail. This is the key to understanding the difference between contagion incomplete and incomplete networks. When the network is complete, banks inthe troubled region have direct claims on banks in every other region. Everyregion takes a small hit (liquidates a small amount of the long asset) and thereis no need for a global crisis. When the network is incomplete, banks in thetroubled region have a direct claim only on the banks in adjacent regions.The banks in other regions are not required to liquidate the long asset untilthey find themselves on the front line of the contagion. At that point, it is toolate to save themselves.

10.1 LIQUIDITY PREFERENCE

In this section we use the standard elements to model liquidity risk. There arethree dates t = 0, 1, 2. There is a single good. This good can be consumedor invested in assets to produce future consumption. There are two types


t = 0 1

1

1

1 1

2

Short liquid asset: (storage)

Long Illiquid asset: R = 1.5

r = 0.4(liquidate)

Figure 10.1. The short and long assets.

of assets, a short asset and a long asset as shown in Figure 10.1. The shortasset is represented by a storage technology. One unit of the consumptiongood invested in the storage technology at date t produces one unit of theconsumption good at date t + 1. Investment in the long asset can only takeplace in the first period and one unit of the consumption good invested in thelong asset at the first date produces R > 1 units of output at the final date.

Each unit of the long asset can be prematurely liquidated to produce0 < r < 1 units of the consumption good at the middle date. Here we assumethat liquidation takes the form of physical depreciation of the asset and the liq-uidation value is treated as a technological constant, the “scrap value.” As wehave indicated in previous chapters, in practice, it is more likely that assets areliquidated by being sold, in which case the liquidation value is determined bythe market price. Introducing a secondary market on which assets can be soldwould complicate the analysis without changing the qualitative features of themodel.

The economy is divided into four ex ante identical regions, labeled A,B,C ,and D. The regional structure is a spatial metaphor that can be interpreted ina variety of ways. The important thing for the analysis is that different regionsreceive different liquidity shocks. Any story that motivates different shocks fordifferent (groups of) banks is a possible interpretationof the regional structure.So a region can correspond to a single bank, a geographical region within acountry, or an entire country; it can also correspond to a specialized sectorwithin the banking industry.

Each region contains a continuum of ex ante identical consumers (depos-itors). A consumer has an endowment equal to 1 unit of the consumptiongood at date 0 and 0 at dates 1 and 2. Consumers are assumed to have theusual preferences: with probability λ they are early consumers and only valueconsumption at date 1; with probability (1 − λ) they are late consumers and

10.1 Liquidity Preference 265

only value consumption at date 2. Then the preferences of the individualconsumer are given by

U (c1, c2) ={u(c1) with probability λ

u(c2) with probability 1 − λ

where ct denotes consumption at date t = 1, 2. The period utility functionsu(·) are assumed to be twice continuously differentiable, increasing and strictlyconcave. In the example we consider to illustrate the model we have

u(·) = ln(ct ).

The probability λ varies from region to region. Let λi denote the probabilityof being an early consumer in region i. There are two possible values of λi ,a high value and a low value, denoted λH and λL , where 0 < λL < λH < 1.The realization of these randomvariables depends on the state of nature. Thereare two equally likely states S1 and S2 and the corresponding realizations ofthe liquidity preference shocks are given in Table 10.1. Note that ex ante eachregion has the same probability of having a high liquidity preference shock.Also, the aggregate demand for liquidity is the same in each state: half theregions have high liquidity preference and half have low liquidity preference.At date 0 the probability of being an early or late consumer is λ = (λH +λL)/2in each region.

All uncertainty is resolved at date 1 when the state of nature S1 or S2 isrevealed and each consumer learns whether he is an early or late consumer. Asusual a consumer’s type is not observable, so late consumers can always imitateearly consumers.

Before introducing the banking sector, it will be convenient to characterizethe optimal allocation of risk.

Table 10.1. Regional liquidity shocks.

A B C D

S1 λH = 0.75 λL = 0.25 λH = 0.75 λL = 0.25S2 λL = 0.25 λH = 0.75 λL = 0.25 λH = 0.75

prob. S1 = prob. S2 = 0.5

Average proportion of early consumers= Average proportion of late consumers= 0.5


10.2 OPTIMAL RISK SHARING

In this section we characterize optimal risk sharing as the solution to a plan-ning problem. Since consumers are ex ante identical, it is natural to treatconsumers symmetrically. For this reason, the planner is assumed to make allthe investment and consumption decisions to maximize the unweighted sumof consumers’ expected utility.

We begin by describing the planner’s problemunder the assumption that theplanner can identify early and late consumers. The symmetry and concavityof the objective function and the convexity of the constraints simplifies theproblem considerably.

• Since there is no aggregate uncertainty, the optimal consumption allocationwill be independent of the state.

• Since the consumers in one region are ex ante identical to consumers inanother region, all consumers will be treated alike.

Without loss of generality, then, we can assume that every early consumerreceives consumption c1 and every late consumer receives c2, independently ofthe region and state of nature. At the first date, the planner chooses a portfolio(y , x) ≥ 0 subject to the feasibility constraint

y + x ≤ 1, (10.1)

where y and x = 1 − y are the per capita amounts invested in the short andlong assets respectively.

• Since the total amount of consumption provided in each period is a constant,it is optimal to provide for consumption at date 1 by holding the short assetand to provide for consumption at date 2 by holding the long asset.

Since the average fraction of early consumers is denoted by λ = (λH + λL)/2,then the feasibility constraint at date 1 is

λc1 ≤ y (10.2)

and the feasibility constraint at date 2 is

(1 − λ)c2 ≤ Rx . (10.3)

At date 0 each consumer has an equal probability of being an early or a lateconsumer, so the ex ante expected utility is

λu(c1) + (1 − λ)u(c2) (10.4)

10.2 Optimal Risk Sharing 267

and this is what the planner seeks tomaximize, subject to the constraints (10.1),(10.2), and (10.3). The unique solution to this unconstrained problem is thefirst-best allocation.

The first-best allocation must satisfy the first-order condition

u′(c1) ≥ u′(c2).

Otherwise, the objective function could be increased by using the short assetto shift some consumption from early to late consumers. Thus, the first-bestallocation automatically satisfies the incentive constraint

c1 ≤ c2, (10.5)

which says that late consumers find it weakly optimal to reveal their true type,rather than pretend to be early consumers. The incentive-efficient allocationmaximizes the objective function (10.4) subject to the feasibility constraints(10.1), (10.2), and (10.3), and the incentive constraint (10.5). What we haveshown is that the incentive-efficient allocation is the same as the first-bestallocation.

Proposition 1 The first-best allocation (x , y , c1, c2) is equivalent to theincentive-efficient allocation, so the first best can be achieved even if theplanner cannot observe the consumers’ types.

Example 1 To illustrate the optimal allocation, suppose that the asset returnsareR = 1.5 and r = 0.4 and the liquidity shocks are λH = 0.75 and λL = 0.25.The average proportion of early consumers is λ = 0.5. The planner chooses yto maximize

0.5 ln( y0.5

)+ 0.5 ln

(R(1 − y)

0.5

).

The first-order condition simplifies to y = 1 − y with solution y = 0.5. Thisgives us the optimal consumption profile (c1, c2) = (1, 1.5).

It is helpful to consider how the planner would achieve this allocation ofconsumption in the context of the example, as shown in Figure 10.2. Sincethere are 0.5 early consumers and 0.5 late consumers there are 0.5 units ofconsumption at date 1 and 0.75 units of consumption needed at date 1 intotal in each region. In order to achieve this first-best allocation, the plannerhas to transfer resources among the different regions. In state S1, for example,there are 0.75 early consumers in regions A and C and 0.25 early consumersin regions B andD. Each region has 0.5 units of the short asset, which provide0.5 units of consumption. So regions A and C each have an excess demand for


A B C DDate 1:

Liquidity demand: 0.75 0.25 0.75 0.25

Liquidity supply: 0.5 0.5 0.5 0.5

Transfer

Date 2:

Liquidity demand: 0.25�1.5 = 0.375

0.5�1.5 = 0.75

0.75�1.5 = 1.125 1.1250.375

Liquidity supply: 0.75 0.750.75

Transfer

0.25 0.25

0.375 0.375

Figure 10.2. Achieving the first-best allocation in state S1.

0.25 units of consumption and regions B and D each have an excess supplyof 0.25 units of consumption. By reallocating this consumption, the plannercan satisfy every region’s needs. At date 2, the transfers flow in the oppositedirection, because regions B and D have an excess demand of 0.375 units eachand regions A and C have an excess supply of 0.375 units each.

10.3 DECENTRALIZATION

In this section we describe how the first-best allocation can be decentralizedby a competitive banking sector. There are two reasons for focusing on thefirst best. One is technical: it turns out that it is much easier to characterizethe equilibrium conditions when the allocation is the first best. The secondreason is that, as usual, we are interested in knowing under what conditionsthemarket“works.”For themoment,we are only concernedwith the feasibilityof decentralization.

The role of banks is to make investments on behalf of consumers and toinsure them against liquidity shocks. We assume that only banks invest in thelong asset. This gives the bank two advantages over consumers. First, the bankscan hold a portfolio consisting of both types of assets, which will typically bepreferred to a portfolio consisting of the short asset alone. Second, by poolingthe assets of a large number of consumers, the bank can offer insurance to con-sumers against their uncertain liquidity demands, giving the early consumerssome of the benefits of the high-yielding long asset without subjecting themto the high costs of liquidating the long asset prematurely at the second date.

10.3 Decentralization 269

In each region there is a continuum of identical banks. We focus on asymmetric equilibrium in which all banks adopt the same behavior. Thus,we can describe the decentralized allocation in terms of the behavior of arepresentative bank in each region.

Without loss of generality, we can assume that each consumer depositshis endowment of one unit of the consumption good in the representativebank in his region. The bank invests the deposit in a portfolio (yi , xi) ≥ 0and, in exchange, offers a deposit contract (c i1, c

i2) that allows the depositor to

withdraw either c i1 units of consumption at date 1 or c i2 units of consumptionat date 2. Note that the deposit contract is not contingent on the liquidity shockin region i. In order to achieve the first best through a decentralized bankingsector, we put (yi , xi) = (y , x) and (c i1, c

i2) = (c1, c2), where (y , x , c1, c2) is the

first-best allocation.The problem with this approach is that, while the investment portfolio sat-

isfies the bank’s budget constraint y + x ≤ 1 at date 1, it will not satisfythe budget constraint at date 2. The planner can move consumption betweenregions, so he only needs to satisfy the average constraint λc1 ≤ y . The repre-sentative bank, on the other hand, has to face the possibility that the fractionof early consumers in its region may be above average, λH > λ, in which caseit will need more than y to satisfy the demands of the early consumers. Itcan meet this excess demand by liquidating some of the long asset, but thenit will not have enough consumption to meet the demands of the late con-sumers at date 2. In fact, if r is small enough, the bank may not be able topay the late consumers even c1. Then the late consumers will prefer to with-draw at date 1 and store the consumption good until date 2, thus causing abank run.

There is no overall shortage of liquidity, it is just badly distributed. One wayto allow the banks to overcome the maldistribution of liquidity is by introdu-cing interbank deposits. Using the data from Example 1, we first consider thelogistics of implementing the first best using interbank deposits when there isa complete network of interbank relationships. Then we consider the case ofan incomplete, connected network.

Example 2 (A complete network) Suppose that the interbank network iscomplete and that banks are allowed to exchange deposits at the first date. Thiscase is illustrated in Figure 10.3. Each region is negatively correlated with twoother regions. The payoffs on these deposits are the same as for consumers. Foreach 1 unit deposited at date 0, the bank can withdraw 1 at date 1 or 1.5 at date2.We are interested in seeing how the first-best allocation can be implemented.Suppose each bank holds a portfolio (y , x) of (0.5, 0.5). If every bank in regioni holds z i = (λH − λ)/2 = 0.25/2 = 0.125 deposits in each of the regions


A

D C

B

Figure 10.3. Complete network.

j �= i, they will be able to supply their depositors with the first-best allocationno matter whether state S1 or S2 occurs. At date 1 the state of nature S isobserved and the banks have to adjust their portfolios to satisfy their budgetconstraints. If the region has a high demand for liquidity, λi = λH = 0.75, itliquidates all of its deposits in other regions. On the other hand, if it has a lowdemand for liquidity, λi = λL = 0.25, it retains the deposits it holds in theother regions until the final date.

Suppose that the state S1 occurs and consider the budget constraint of abank in region A with a high demand for liquidity as shown in Figure 10.4.The first thing to notice is that its deposit of 0.125 with a bank in region C ,which also has high demand, cancels with the claim on it of 0.125 from regionC . In addition it must pay c1 = 1 to the fraction λH = 0.75 of early consumersin its own region for a total of 0.75. On the other side of the ledger, it hasy = 0.5 units of the short asset and claims to 2 × 0.125 = 0.25 deposits inregions B and D. Thus, it has enough assets to meet its liabilities. The sameanalysis holds for a bank in region C .

Next consider a bank in region B, which has low liquidity demand. It mustpay c1 = 1 to a fraction λL = 0.25 of their own depositors and redeem2 × 0.125 = 0.25 deposits from the banks in the regions A and C with highliquidity demand. It has y = 0.5 units of the short asset tomeet these demands,so its assets cover its liabilities. A similar analysis holds for a bank in regionD.

At date 2,all the banks liquidate their remaining assets and it canbe seen fromFigure 10.4 that they have sufficient assets to cover their liabilities. Consider abank in region A or C first. It must pay c2 = 1.5 to the fraction 1− λH = 0.25of early consumers in its own region for a total of 0.375. The banks in regionsB and D have total claims of 2 × 0.125 × 1.5 = 0.375. The total claims on itare 0.75. On the other side of the ledger, it has 1 − y = 0.5 units of the longasset that gives a payoff of 0.5× 1.5 = 0.75. Thus, it has enough assets to meetits liabilities. The same analysis holds for a bank in region C .

Banks in regions B andDmust each pay c2 = 1.5 to a fraction 1−λL = 0.75of their own depositors so their total liabilities are 0.75 × 1.5 = 1.125.In terms of their assets they have 0.5 of the long asset which has a pay-off of 0.5 × 1.5 = 0.75 and deposits in banks in regions A and C worth


Date 1:

Date 2:

A

D C

B

Cross deposits cancel

Demand = 0.75 Supply = 0.5

0.125

0.125

0.125

0.125

Demand = 0.75Supply = 0.5



A

D C

B

Cross deposits cancel


0.1875

0.1875

0.1875

0.1875




Figure 10.4. The flows between banks in state S1 with a complete network structure.

2 × 0.125 × 1.5 = 0.375. So its total assets are 0.75 + 0.375 = 1.125 andthese cover its liabilities.

Thus, by shuffling deposits among the different regions using the interbanknetwork, it is possible for banks to satisfy their budget constraints in eachstate S and at each date t = 0, 1, 2 while providing their depositors with thefirst-best consumption allocation through a standard deposit contract.

Example 3 (An incomplete network) The interbank network in the preced-ing section is complete in the sense that a bank in region i can hold deposits inevery other region j �= i. In some cases, this may not be realistic. The bankingsector is interconnected in a variety of ways, but transaction and informationcosts may prevent banks from acquiring claims on banks in remote regions. Tothe extent that banks specialize in particular areas of business or have closerconnections with banks that operate in the same geographical or political unit,deposits may tend to be concentrated in “neighboring” banks. To capture thiseffect,which is crucial in the sequel,we introduce the notion of incompletenessof the interbank network by assuming that banks in region i are allowed tohold deposits in some but not all of the other regions. For concreteness, weassume that banks in each region hold deposits only in one adjacent region,


as shown in Figure 10.5. It can be seen that banks in regionA can hold depositsin region B, banks in region B can hold deposits in region C and so on.

This network structure again allows banks to implement the first-best allo-cation. The main difference between this case and the previous one is thatinstead of depositing 0.125 in two banks, each bank deposits 0.25 in one bank.The transfers at dates 1 and 2 are then as shown in Figure 10.6.

One interesting feature of the network structure in Figure 10.5 is that,although each region is relying on just its neighbor for liquidity, the entireeconomy is connected. Region A holds deposits in region B, which holds

A

D C

B

Figure 10.5. Incomplete network structure.

Date 1:

Date 2:

A

D C

BDemand = 0.75 Supply = 0.5

0.25

0.25




A

D C


0.375

0.375




Figure 10.6. The flows between banks in state S1 with an incomplete networkstructure.


deposits in region C , and so on. In fact, this is unavoidable given the net-work structure assumed. Consider the alternative network structure shown inFigure 10.7. Region A holds deposits in region B and region B holds depositsin region A. Likewise, region C holds one unit of deposits in region D andregion D holds one unit of deposits in region C . This network structure ismore incomplete than the one in Figure 10.2 and the pattern of holdings inFigure 10.5 is incompatible with it. However, it is possible to achieve the firstbest through the pattern of holdings in Figure 10.8. This is true even though

A

D C

B

Figure 10.7. A separated incomplete network structure.

Date 1:

Date 2:

A

D C


0.25

0.25




A

D C


0.375

0.375




Figure 10.8. The flows between banks in state S1 with a separated incomplete networkstructure.


the economy is disconnected, since regions A and B trade with each other butnot with regions C and D and regions C and D trade with each other but notwith regions A and B. Again, these patterns do not seem to have any signifi-cance as far as achieving the first best is concerned; but they turn out to havestriking differences for contagion.

10.4 CONTAGION

To illustrate how a small shock can have a large effect, we use the decentral-ization results from Section 10.3. Then we perturb the model to allow for theoccurrence of a state S in which the aggregate demand for liquidity is greaterthan the system’s ability to supply liquidity and show that this can lead to aneconomy-wide crisis.

The network structure is assumed to be given by Figure 10.5. The cor-responding allocation requires each bank to hold an initial portfolio ofinvestments (y , x) and offer a deposit contract (c1, c2), where (y , x , c1, c2) isthe first-best allocation. In order to make this deposit contract feasible, therepresentative bank in each region holds z = 0.25 deposits in the adjacentregion. Note that z is the minimal amount that is needed to satisfy the bud-get constraints. It will become apparent below that larger cross holdings ofdeposits, while consistent with the first best in Section 10.3, would make thecontagion problem worse.

Now, let us take the allocation as given and consider what happens whenwe “perturb” the model. By a perturbation we mean the realization of a stateS that was assigned zero probability at date 0 and has a demand for liquiditythat is very close to that of the states that do occur with positive probability.Specifically, the liquidity shocks are shown inTable 10.2. In state S, every regionhas the previous average demand for liquidity λ except for region A where thedemand for liquidity is somewhat higher λ + ε. The important fact is that theaverage demand for liquidity across all four regions is slightly higher than inthe normal states S1 and S2. Since the abnormal state S occurs with negligibleprobability (in the limit, probability zero) it will not change the allocation at

Table 10.2. Regional liquidity shocks with perturbation.

A B C D

S1 λH = 0.75 λL = 0.25 λH = 0.75 λL = 0.25S2 λL = 0.25 λH = 0.75 λL = 0.25 λH = 0.75

S λ + ε = 0.5 + ε λ = 0.5 λ = 0.5 λ = 0.5

10.4 Contagion 275

date 0. In states S1 and S2 the continuation equilibrium will be the same asbefore at date 1; in state S the continuation equilibrium will be different.

In the continuation equilibrium beginning at date 1, consumers will opti-mally decide whether to withdraw their deposits at date 1 or date 2 and bankswill liquidate their assets in an attempt to meet the demands of their deposit-ors. Early consumers always withdraw at date 1; late consumers will withdrawat date 1 or date 2 depending on which gives them the larger amount of con-sumption. Because we want to focus on essential bank crises, we assume thatlate consumers will always withdraw their deposits at date 2 if it is (weakly)optimal for them to do so. Banks are required to meet their promise to pay c1units of consumption to each depositor who demands withdrawal at date 1. Ifthey cannot do so, they must liquidate all of their assets at date 1. The proceedsof the liquidation are split pro rata among depositors in the usual way. If thebank canmeet its obligations at date 1, then the remaining assets are liquidatedat date 2 and given to the depositors who have waited until date 2 to withdraw.In the rest of this section, we describe the continuation equilibrium at date 1in state S, assuming the actions consistent with the first best at date 0.

10.4.1 The liquidation “pecking order”

At date 1 a bank can find itself in one of three conditions. A bank is said to besolvent, if it can meet the demands of every depositor who wants to withdraw(including banks in other regions) by using only its liquid assets, that is, theshort asset and the deposits in other regions. The bank is said to be insolvent ifit canmeet the demands of its deposits but only by liquidating some of the longasset. Finally, the bank is said to be bankrupt if it cannot meet the demands ofits depositors by liquidating all its assets.

These definitions are motivated by the assumption that banks will alwaysfind it preferable to liquidate assets in a particular order at date 1. We call thisthe “pecking order” for liquidating assets and it goes as follows: first, the bankliquidates the short asset, then it liquidates deposits, and finally it liquidates thelong asset. To ensure that the long asset is liquidated last, we need an additionalassumption,

R

r>c2c1

(10.6)

which is maintained in the sequel. Since the first-best consumption allocation(c1, c2) is independent of r (this variable does not appear in the first-bestproblem in Section 10.2) we can always ensure that condition (10.6) is satisfied


by choosing r sufficiently small. It can be seen that (10.6) is satisfied in ourexample since R/r = 1.5/0.5 > 1.5/1 = c2/c1.

Each of the three assets offers a different cost of obtaining current (date 1)consumption in terms of future (date 2) consumption. The cheapest is theshort asset. One unit of the short asset is worth one unit of consumption todayand, if reinvested in the short asset, this is worth one unit of consumptiontomorrow. So the cost of obtaining liquidity by liquidating the short asset is1. Similarly, by liquidating one unit of deposits, the bank gives up c2 unitsof future consumption and obtains c1 units of present consumption. So thecost of obtaining liquidity by liquidating deposits is c2/c1. From the first-ordercondition u′(c1) = Ru′(c2), we know that c2/c1 > 1. Finally, by liquidatingone unit of the long asset, the bank gives up R units of future consumptionand obtains r units of present consumption. So the cost of obtaining liquidityby liquidating the long asset is R/r . Thus, we have derived the pecking order,short assets, deposits, long assets:

1 <c2c1

<R

r.

In order to maximize the interests of depositors, the bank must liquidate theshort asset before it liquidates deposits in other regions before it liquidates thelong asset.

The preceding argument assumes that the bank in which the deposit is heldis not bankrupt. Bankruptcy requires that all assets of the bankrupt institutionbe liquidated immediately and the proceeds distributed to the depositors. Sothe preceding analysis only applies to deposits in non-bankrupt banks.

10.4.2 Liquidation values

The value of a deposit at date 1 is c1 if the bank is not bankrupt and it is equalto the liquidation value of all the bank’s assets if the bank is bankrupt. Letqi denote the value of the representative bank’s deposits in region i at date 1.If qi < c1 then all the depositors will withdraw as much as they can at date1. In particular, the banks in other regions will be seeking to withdraw theirclaims on the bank at the same time that the bank is trying to redeem its claimson them. All depositors must be treated equally, that is, every depositor gets qi

from the bank for each unit invested at the first date, whether the depositoris a consumer or a bank from another region. Then the values of qi must bedetermined simultaneously. Consider the representative bank in region A, forexample. If all the depositors withdraw, the total demands will be 1 + z , since

10.4 Contagion 277

the banks in region D hold z deposits and the consumers in region A hold 1deposit. The liabilities of the bank are valued at (1 + z)qA . The assets consistof y units of the short asset, x units of the long asset, and z deposits in regionB. The assets are valued at y + rx + zqB . The equilibrium values of qA and qB

must equate the value of assets and liabilities

qA = y + rx + zqB(1 + z) . (10.7)

A similar equation must hold for any region i in which qi < c1.If qB = c1 then we can use this equation to calculate the value of qA ; but

if qB < c1 then we need another equation to determine qB and this equationwill include the value of qC , and so on.

10.4.3 Buffers and bank runs

Abank canmeet a certain excess demand in liquidity at date 1,whichwe call thebuffer, by liquidating the illiquid long-term asset before a run is precipitated.To see how this works consider the following examples of what happens in Swhen there is an incomplete network structure as in Figure 10.5.

Example 4 ε = 0.04In this case it follows from Table 10.2 that the proportions of early and lateconsumers are as follows.

Early consumers Late consumers

Proportion of Bank A customers 0.54 0.46

Since the bank has promised c1 = 1 itmust comeupwith 0.54 of consumption.To do this it goes through its pecking order.

(i) To meet the first 0.5 of its liquidity needs of 0.54, it liquidates its 0.5holding of the short asset.

(ii) It needs more liquidity so next it calls in its deposit from Bank B. NowfromTable 10.2 it can be seen that in state S BankB has a liquidity demandof 0.5 from its early consumers. It uses its proceeds from the short assetof 0.5 to cover this. When Bank A calls in its deposit then Bank B goes tothe second item on its pecking order and calls in its deposit from BankC . Bank C is in a similar position to Bank B and calls in its deposit fromBank D, which in turn calls in its deposit from Bank A. So overall Bank


A is no better off in terms of raising liquidity. It has an extra 0.25 of theliquid asset but it also has an extra demand for liquidity from Bank Dof 0.25.

(iii) Bank A goes to the third item in its pecking order, which is to liquidate itsholding of the long asset. Since r = 0.4 and it needs 0.04 it must liquidate0.04/0.4 = 0.1.

The use of the short-term asset and the liquidation of 0.1 of its long-termasset means that Bank A is able to meet its liabilities. It has 0.5 − 0.1 = 0.4of the long asset remaining. At date 2 it will have 0.4 × 1.5 = 0.6 from thisholding of the long asset. It can distribute c2 = 0.6/0.46 = 1.30 to the lateconsumers. Since this is above the c1 = 1 they would obtain if they pretendedto be early consumers, there is no run on the bank. The only effect of theincreased liquidity demand is a reduction in the consumption of Bank A’s lateconsumers as it is forced to liquidate some of its buffer.

Example 5 ε = 0.10Next consider what happens if the liquidity shock is larger. Now theproportions of early and late consumers are as follows.

Early consumers Late consumers

Proportion of Bank A customers 0.6 0.4

The sequence of events is the same as before. Bank A goes through its peckingorder but now it needs an extra 0.1 of liquidity. It must therefore liquidate0.1/0.4 = 0.25 of the long asset. At date 2 it would have 0.5 − 0.25 = 0.25 ofthe long asset, which would generate a payoff of 0.25 × 1.5 = 0.375. It coulddistribute c2 = 0.375/0.4 = 0.94 to the late consumers. Since this is below thec1 = 1 they would obtain if they pretended to be early consumers, there is arun on the bank.

In the run on Bank A all depositors, including Bank D, withdraw and takea loss. There is a spillover to Bank D. The key question is whether there iscontagion so that Bank D also goes bankrupt.

Assume initially that there is no contagion and in particular that BankA’s deposit claim on Bank B is worth 0.25. We shall check shortly whether thisassumption is consistent with what everybody receives. Using (10.7) it can beseen that the pro rata claim on Bank A is

qA = 0.5 + 0.5 × 0.4 + 0.25

1.25= 0.76.

10.4 Contagion 279

This means that Bank D’s 0.25 deposit claim is worth 0.76 × 0.25 = 0.19.They have claims of 0.5 from their depositors and also of 0.25 from Bank C ’sdeposit claim on them for a total of 0.75. Since their total liquidity from theirholdings of the short asset and their deposit in Bank A is 0.5 + 0.19 = 0.69they need to draw on their buffer and liquidate enough of their long-term assetto give them 0.75 − 0.69 = 0.06. The total amount they must liquidate of thelong asset is 0.06/0.4 = 0.15. Bank D thus has 0.5 − 0.15 = 0.35 of the longasset remaining. At date 2 it will have 0.35 × 1.5 = 0.525 from this holding ofthe long asset. It can distribute c2 = 0.525/0.5 = 1.05 to the late consumers.Since this is above the c1 = 1 they would obtain if they pretended to be earlyconsumers, there is no run on Bank D and no contagion.

Note that even if ε was larger, there would still be no contagion. Bank A isbankrupt and all its depositors irrespective of whether they are early or lateconsumers get 0.76. In order for there to be contagion we need R to be lowerthan 1.5 so that there is a run on Bank D. This is the case we consider next.

Example 6 ε = 0.10, R = 1.2Example 6 is the same as Example 5 except for the last step. At date 2, insteadof having 0.35 × 1.5 = 0.525 from their holding of the long asset, Bank D willhave 0.35 × 1.2 = 0.42. It could distribute c2 = 0.42/0.5 = 0.84 to the lateconsumers. Since this is below the c1 = 1 they perceive they would obtain ifthey pretended to be early consumers, there is a run on BankD and now thereis contagion. The analysis for Banks C and B is just the same as for Bank Dand so all banks go bankrupt.

In the contagion equilibrium all banks go down and are liquidated so qi = qfor i = A,B,C ,D. Using (10.7) we have,

q = 0.5 + 0.5 × 0.4 + 0.25 × q1.25

.

Solving for q gives

q = 0.7.

So all depositors obtain 0.7 and welfare is lower than when there is nocontagion.

It can be seen that there is always contagion provided R is sufficiently low.In fact, there will be contagion whenever 0.35 × R < 0.5 or, equivalently,R < 1.43.

Proposition 2 Consider the model with network structure described inFigure 10.5 and perturb it by the addition of the zero probability state S.Suppose that each bank chooses an investment portfolio (y , x , z) and offers


a deposit contract (c1, c2), where (x , y) is the first-best investment portfolio,(c1, c2) is the first-best consumption allocation, and z = (λH − λ). If the liq-uidity shock ε > 0 is sufficiently large andR > 1 is sufficiently low, then, in anycontinuation equilibrium, the banks in all regionsmust go bankrupt at date 1 instate S.

We have illustrated this proposition for a numerical example. Amore formalversion of the proposition is provided in Allen and Gale (2000).

10.4.4 Many regions

So farwe have only considered the case of four regionsA,B,C , andD. However,it can be seen from the final step of the example above that there wouldbe contagion no matter how many regions there are. When Bank D goesbankrupt all the other regions also go bankrupt. The same argument wouldhold with a thousand or many thousands of regions. Thus, even though theinitial shock only occurs in one region, which can be an arbitrarily small partof the economy, it can nevertheless cause banks in all regions to go bankrupt.This is why contagion can be so damaging.

Proposition 3 Proposition 2 holds no matter how many regions there are.

10.5 ROBUSTNESS

The incompleteness of networks is important for the contagion result. In theexample we considered last where there was contagion when networks wereas in Figure 10.5 it can be shown that if there is a complete network as inFigure 10.3, there is no contagion. The key difference is that now with a com-plete network, each bank deposits 0.125 in two other banks rather than 0.25 injust one bank.

Example 7 ε = 0.10, R = 1.2The analysis is the same as before except when we are considering whetherthere is contagion. Bank B and D’s deposit claims of 0.125 on Bank A are nowworth 0.76 × 0.125 = 0.095. Banks B, C , and D have claims of 0.5 in totalfrom their early consumers and 0.125 from the three banks that have depositsin them. They have liquidity of 0.5 from their short asset and 0.125 from two oftheir deposits and 0.095 from their deposit in Bank A. They need to liquidate

10.6 Containment 281

enough of the long asset to raise 0.125 − 0.095 = 0.03. Given r = 0.4 theamount liquidated is 0.03/0.4 = 0.075. They thus have 0.5 − 0.075 = 0.425of the long asset remaining. At date 2 they will have 0.425 × 1.2 = 0.51 fromthis holding of the long asset. They can distribute c2 = 0.51/0.5 = 1.02 tothe late consumers. Since this is above the c1 = 1 they would obtain if theypretended to be early consumers, there is no run on Banks B,C , andD and nocontagion.

The reason there is no contagion in the case with a complete network isthat the liquidation of assets is spread among more banks so there are morebuffers to absorb the shock. This means that banks do not hit the discontinuityassociated with a run when all the bank’s assets are liquidated at a significantloss and as a result there is no contagion. We have illustrated this for theexample. Allen and Gale (2000) again show that a similar result holds moregenerally. Complete networks are less susceptible to contagion than incompletenetworks.

10.6 CONTAINMENT

The critical ingredient in the example of contagion analyzed in Section 10.4 isthat any two regions are connected by a chain of overlapping bank liabilities.Banks in region A have claims on banks in region B, which in turn have claimson banks in region C , and so on. If we could cut this chain at some point, thecontagion that begins with a small shock in region A would be contained insubset of the set of regions.

Consider the incomplete network structure in Figure 10.7 and the allocationthat implements the first-best allocation for our example, which is shown inFigure 10.8. The allocation requires banks in regions A and B to have claimson each other and banks in regions C and D to have claims on each other, butthere is no connection between the region {A,B} and the region {C ,D}. If stateS occurs, the excess demand for liquidity will cause bankruptcies in region Aand they can spread to region B, but there is no reason why they should spreadany further. Banks in regionsC andD are simply not connected to the troubledbanks in regions A and B.

Comparing the three network structures we have considered so far, com-plete networks in Figure 10.3, incomplete networks in Figure 10.5, and thedisconnected network structure in Figure 10.7, we can see that there is a non-monotonic relationship between completeness or incompleteness of networksand the extent of the financial crisis in state S. With the complete networkstructure of Figure 10.3 the crisis is restricted to region A, with the network


structure in Figure 10.5 the crisis extends to all regions, and with the networkstructure in Figure 10.7 the crisis is restricted to regions A and B.

It could be argued that the network structures are not monotonicallyordered: the complete network does contain the other two, but the paths inthe network in Figure 10.7 are not a subset of the network in Figure 10.5. Thiscould be changed by adding paths to Figure 10.2, but then the equilibrium ofFigure 10.7 would also be an equilibrium of Figure 10.5. This raises an obviousbut important point, that contagion depends on the endogenous pattern offinancial claims. An incomplete network structure like the one in Figure 10.5may preclude a complete pattern of financial connectedness and thus encour-age financial contagion; but a complete network structure does not imply theopposite: even in a complete network there may be an endogenous choice ofoverlapping claims that causes contagion. In fact, the three equilibria consid-ered so far are all consistent with the complete network structure. There areadditional equilibria for the economy with the complete network structure.Like the three considered so far, they achieve the first best in states S1 andS2, but have different degrees of financial fragility in the unexpected state S,depending on the patterns of interregional deposit holding.

What is important about the network structure in Figure 10.5, then, is thatthe pattern of interregional cross holdings of deposits that promotes the pos-sibility of contagion, is the only one consistent with this network structure.Since we are interested in contagion as an essential phenomenon, this net-work structure has a special role. The complete network economy, by contrast,has equilibria with and without contagion and provides a weaker case for thelikelihood of contagion.

10.7 DISCUSSION

The existence of contagion depends on a number of assumptions. The firstis that financial interconnectedness takes the form of ex ante claims signed atdate 0. The interbank loan network is good in that it allows reallocation ofliquidity. But when there is aggregate uncertainty about the level of liquiditydemand this interconnectedness can lead to contagion. It is not importantthat the contracts are deposit claims. The same result holds with contingentor discretionary contracts because the interbank claims net out. Ex ante con-tracts will always just net out. Spillover and contagion occur because of thefall in asset values in adjacent regions, not the form of the contract. Theinterbank network operates quite differently from the retail market in thisrespect.

10.7 Discussion 283

Note that if there is an ex post loan market so contracts are signed at date 1rather than date 0 then there can be the desirable reallocation of liquidity butno contagion. The reason that there will be no contagion is that the interest ratein the ex postmarketmust compensate lenders for the cost of liquidating assetsbut at this rate it will not be worth borrowing. However, there are the usualdifficulties with ex post markets such as adverse selection and the “hold-up”problem. If the long asset has a risky return then ex post markets will also notbe optimal.

We have simplified the problem considerably to retain tractability. In partic-ular, by assuming state S occurs with zero probability, we ensure the behaviorof banks is optimal since the interbank deposits and the resulting allocationremains efficient. When S occurs with positive probability the trade-offs willbe more complex. However, provided the benefits of risk sharing are largeenough interbank deposits should be optimal and this interconnection shouldlead to the (low probability) possibility of contagion. When S occurs withpositive probability a bank can prevent runs by holding more of the liquidasset. There’s a cost to this in states S1 and S2 though. When the probability ofS is small enough this cost is not worth bearing. It is better to just bear the riskof contagion.

The focus in this paper is on financial contagion as an essential featureof equilibrium. We do not rely on arguments involving multiple equilibria.The aim is instead to show that under certain conditions every continu-ation equilibrium at date 1 exhibits financial contagion. Nonetheless, thereare multiple equilibria in the model and if one is so disposed one can use themultiplicity of equilibria to tell a story about financial contagion as a sunspotphenomenon.

For simplicity, we have assumed that the long asset has a non-stochasticreturn, but it would be more realistic to assume that the long asset is risky. Wehave seen at many points in this book, such as Chapter 3 on intermediation,that when the long asset is risky it is negative information about future returnsthat triggers bank runs. In the present framework, uncertainty about long assetreturns could be used both tomotivate interregional cross holdings of depositsand to provoke insolvency or bankruptcy. The results should be similar. Whatis crucial for the results is that the financial interconnectedness between theregions takes the form of claims held by banks in one region on banks inanother region.

We have shown in this chapter that a small shock in a single region or bankcan bring down the entire banking system no matter how big this system isrelative to the shock. What is key for this is the network structure for inter-bank deposits. If there are relatively few channels of interconnectedness thencontagion is more likely. Clearly transaction costs mean it is much too costly


for every bank to hold an account with every other bank so networks are com-plete. However, one low cost equivalent of this is to have a central bank thatis connected with every other bank. The theory here thus provides a rationalefor central banks.

10.8 APPLICATIONS

In this section we try to bridge the gap between the theory developed in thepreceding sections and empirical applications. The model we have presentedhere describes an artificial and highly simplified environment in which it ispossible to exhibit the mechanism by which a small shock in one region can betransmitted to other regions. This result depends on a number of assumptions,some of them quite restrictive, and some of them made in order to keep theanalysis tractable. In any case, the model is quite far removed from the worldin which policymakers have to operate. Nonetheless, although the model onlyprovides an extremely simplified picture of the entire financial system, thebasic ideas can be applied to real data in order to provide some indication ofthe prospects for contagion in actual economies. There have now been severalstudies of this sort, carried out on data from different countries. An excellentsurvey of this literature is contained in Upper (2006). Here we start with oneparticularly transparent example, which will serve to illustrate what can bedone in practice.

10.8.1 Upper andWorms (2004)

In an important study, Upper and Worms (2004), henceforth UW, use data oninterbank deposits to simulate the possibility of contagion among the banksin the German financial system. The UW model is very simple. There is afinite number of banks indexed by i = 1, ..., n. Each bank i has a level ofcapital denoted by the number ci ≥ 0. For any ordered pair of banks

(i, j)

there is a number xij ≥ 0 that denotes the value of the claims of bank ion bank j . These claims may represent deposits in bank j or bank loans tobank j or some combination. The numbers ci and xij represent the data oninterbank relationships and that will be used as the basis for the analysis ofcontagion.

The other crucial parameter that plays a role in the process of contagion isthe loss ratio θ . When a bank goes bankrupt, its assets will be liquidated inorder to pay off the creditors. Typically, the assets will be sold for less than

10.8 Applications 285

their “book” or accounting value. The reasons why assets are sold at firesaleprices are numerous and well known. Some potential buyers, perhaps thosewho value the assets highly, may be unable to bid for them because of a lack ofliquidity. Or it may be that assets are sold in haste, before every potential buyeris ready to bid on them, leading to a non-competitive market for the assetson which buyers can pick up the assets for less than the fair value. There isalso the problem of asymmetric information. If potential buyers have limitedinformation about the quality of the assets (and some bank assets, such asloans, are notoriously hard to value), fear of adverse selection will cause buyersto incorporate a “lemons” discount in their bids. Part of the loss from sellingthe assets at firesale prices will be absorbed by the bank’s capital, but the lossesmay be so great that the liquidated value of the assets is less than the valueof the bank’s liabilities. In that case, part of the losses will be passed on tothe creditors, who only receive a fraction of what they are owed. The lossratio θ measures the fraction of the creditors’ claim that is lost because of theliquidation.

The process of contagion begins with the failure of a single bank. If theloss ratio is positive, all the creditors will lose a positive fraction of the claimsagainst the failing bank. If this loss is big enough, some of the affected banksmay fail. This is the first round of contagion. The failure of these additionalbanks will have similar effects on their creditors, some of whom may fail in thesecond round. These effects continue in round after round as the contagionspreads throughout thefinancial sector. This recursive descriptionof contagionsuggests an algorithm for calculating the extent of contagion from the failureof a single bank.

The first step of the algorithm takes as given the failure of a given bank andcalculates the impact on the creditor banks. Suppose that bank j fails and banki is a creditor of bank j . Bank i’s claim on bank j is xij and a fraction θ of thisis lost, so bank i suffers a loss of θxij . If the loss θxij is less than bank i’s capitalci , the bank can absorb the loss although its capital will be diminished. If theloss is greater than the value of the bank’s capital, however, the bank’s assetsare reduced to the point where they are lower than its liabilities. When assetsare less than liabilities, the bank is insolvent and must declare bankruptcy. Sothe failure of bank j will cause the failure of bank i if and only if the loss θxijis greater than its capital ci :

θxij > ci .

The banks that satisfy this inequality constitute the first round of contagionfrom the failure of bank j .


Let I1 denote the set of banks i that failed in the first round of contagion,plus the original failed bank j . The failure of bank j has spread by contagionto all the banks in the set I1 but the contagion does not stop there. Banks thatdid not fail as a result of the original failure of bank j may now fail because ofthe failure of banks in I1. Suppose that i does not belong to I1. For every j inI1, bank i loses θxij so its total loss is

∑j∈I1 θxij . Then bank i will fail in what

we call the second round if and only if

∑j∈I1

θxij > ci .

Let I2 denote the set of banks that fail in the second round plus all the banksincluded in I1. We can continue in this way calculating round after round offailures until the set of failed banks converges. Let Ik denote the set of banksthat fail in the k-th and earlier rounds. Since there is a finite number of banks,contagion must come to an end after a finite number of periods. It is easy tosee from the definition that if Ik = Ik+1 then Ik = Ik+� for every � > 0.

The interesting question is how far the contagion will go. There are variousways of measuring the extent of contagion. UW report various measures,including the total number of failed banks and the percentage of assets infailed banks. The procedure for estimating the extent of contagion is as follows.Choose an arbitrary bank i and assume that it fails, calculate the set of banksthat will fail as a result of the failure of bank i (the procedure described above),and then repeat this procedure for every possible choice of the starting banki. The extent of contagion is the least upper bound of the relevant measure,say, the total number of failed banks or the percentage of assets in failed banks,over all starting values i. In other words, we choose the initial failed bank tomaximize the measure of contagion and call that maximized value the extentof contagion.

The data required for this exercise are not available in completely disag-gregated form so UW were forced to approximate it using partially aggregateddata. German banks are required to submitmonthly balance sheets to the Bun-desbank in which they report interbank loans and deposits classified accordingto whether their counterparty is a foreign or domestic bank, a building society,or the Bundesbank. Savings banks and cooperative banks have to report inaddition whether the counterparty is a giro institution or a cooperative centralbank. All banks are also required to divide their lending into five maturity cat-egories. What are not reported are the actual bilateral positions between pairsof banks. This information has to be interpolated from the aggregates reportedby individual banks. Nonetheless, the division of interbank loans and deposits


into so many categories allows UW to construct a number of matrices cor-responding to lending between different types of banks in different maturityclasses. Since we are ultimately concerned only with the aggregate amount ofbilateral lending, these calculations would be of no interest if we could observebilateral exposures directly. Since we cannot observe bilateral exposures, theadditional information on the breakdown of lending by individual banks isuseful in estimating the unobserved bilateral exposures. UW use a complex,recursive algorithmwhich uses the sumof lending and borrowing by each bankin each of several categories (defined by type of counterparty and maturity)to estimate the bilateral exposures for those categories. By applying this pro-cedure to the 25 separate matrices corresponding to lending between differenttypes of banks and in different maturities, UW are able to get more preciseestimates of the true bilateral exposures, because many banks are active onlyin some of these categories. The resulting matrices are then summed to giveaggregate bilateral exposures. Since they only have data on domestic banks anddomestic branches of foreign banks,UW eliminate exposures to foreign banks,building societies and the Bundesbank. They are left with a closed system inwhich assets and liabilities sum to zero.

The “full information”method shows that the patterns of lending and bor-rowing are quite different for different classes of banks. Among their broadconclusions about the structure of lending are the following.

• Banks tend to have larger claims on banks in the same category. For example,commercial banks transact much more with other commercial banks thanone would expect from the baseline matrix.

• Similarly, the head institutions of the two giro systems (cooperative banksand savings banks) have a large proportion of the loans and deposits inthe individual banks in each giro system. There are almost no deposits heldbetween individual banks at the base level of the same giro system.

• There are therefore two tiers to the banking system, the lower tier consistingof savings and cooperative banks, the upper tier consisting of commercialbanks, the head banks of the giro systems (Landesbanken and coopera-tive central banks) plus a variety of other banks. Whereas the lower tierbanks have little exposure to banks in the same tier, the upper tier bankshave transactions with a variety of other banks including banks in othercategories.

The two-tier system falls between the complete network and the incompletenetwork emphasized in the theory model. As a network it is incomplete butthe hubs in the upper tier play an important role in integrating the system.In addition, the theoretical model assumes that all banks are ex ante identical,


bank A

bank D

bank D3bank D1 bank D2 bank C1 bank C3

bank C

bank C2Lower

tier

Uppertier

bank B

Figure 10.9. Two-tier structure of German interbank holdings (Figure 4 in Upper andWorms 2004).

whereas the key to understanding the two tier system in Germany is the dif-ference in sizes and specialized functions of the banks in each tier. A stylizedpicture of the two-tier system, reproduced from UW, is shown in Figure 10.9.

As mentioned above, the loss ratio is a key parameter in determining thepossibility of contagion. Since data on loss ratios applicable to the Germanbanking system are not available, UW consider a range of values of θ andcalculate the incidence of contagion, as measured by a variety of statistics, foreach value of θ . Recall that we are referring to the maximum incidence here,that is, the maximum extent of contagion associated with the failure of a singlebank.We reproduce in Figure 10.10 the relationship between the loss ratio andboth the maximum number of banks and the maximum percentage of totalassets affected using the full information matrix of bilateral exposures.

The relationship between the loss ratio and the extent of contagion hasseveral important features that can be seen in Figure 10.10 or in the underlyingcalculations.

• There is always contagion for any loss ratio. In fact, there are 17 banks thatfail in the first round, independently of which bank is chosen to fail first.These are all small banks and the assumptions on which the interpolationof interbank exposures is based may be unrealistic in their case.

• There appears to be a critical value of θ (around 40%) at which the extentof contagion resulting from a single bank failure increases very sharply.

• For very large values of θ (possibly unrealistically large values), the contagionextends to most of the financial system.


0.2 0.4 0.6 0.80

1000

2000

3000Number of banks affected

0.2 0.4 0.6 0.80

20

40

60

80

100% of total assets affected

Loss ratio u

Maximum effect

Average effect

Maximum effect

Average effect

Loss ratio u

Figure 10.10. Loss ratio and the severity of contagion in the absence of a “safety net”(Figure 5 in Upper and Worms 2004).

UWprovide a number of other results, relating to the dynamics of contagionand the disparate impact of contagion on different types of banks.

The UW methodology provides a method for estimating the incidence ofcontagion using hypothetical values of the loss ratio and estimates of bilateralexposures based on available data. It provides a quantitative assessment of thefinancial fragility of the banking system aswell as some interesting insights intothe sensitivity of the results to different structural parameters of the system.At the same time, the approach has a number of limitations.

• UW focus on interbank holdings, but there are other sources of instability,for example, shocks originating outside the financial system, that may leadto contagion.

• UW interpret their algorithm for calculating the extent of contagion as adynamic process in which each round can be thought of as a different timeperiod. In a truly dynamic model, banks would be able to change theirportfolios in each period as the contagion progressed. It is not clear how


this would affect the analysis. On the one hand, banks might be able to takedefensive action to protect themselves against contagion. On the other hand,each bank’s attempt to defend itself, for example, by withdrawing funds fromother banks, may actually accelerate the process of contagion.

• A related point is that the analysis assumes that asset prices and interest ratesremain constant throughout the process. If large scale liquidation of assets istaking place (or is anticipated), there may be a strong impact on asset prices.A fall in asset prices may increase the vulnerability of banks by reducingtheir capital. Again, this may accelerate the process of contagion.

10.8.2 Degryse and Nguyen (2004)

Another interesting study in this vein has been carried out by Degryse andNguyen (2004), henceforth DN, for the Belgian banking system. The mostinteresting feature of DN is that it uses data on interbank loans and depositsfor the period 1993–2002. This allows the authors to study the evolution of therisk of contagion over a ten year period. In the years between 1998 and 2001,the banking system experienced substantial consolidation which changed thestructure of the industry as well as interbank exposures. At the beginning ofthe period 1993–2002, the Belgian banking system could be characterized as acomplete network, in which all banks have more or less symmetric exposures.By the end of the period, it resembled an incomplete network with multiplemoney centers, in which themoney center banks have symmetric links to otherbanks and the non-money center banks do not have links with each other.

DN simulate the risk of contagion at the beginning and end of the period1993–2002 using a variety of values of loss given default (LGD), which is theircounterpart to the loss ratio θ in UW. They find that the risk of contagionhas fallen over the period and is quite low by the end. Even with an unrealis-tically high LGD of 100%, they calculate that banks representing less than fivepercent of total assets would be affected by contagion following the failure ofa single Belgian bank. Thus, the banking system has become less completeover the period during which the risk of contagion has fallen. This standsin contrast to the result of Allen and Gale (2000), which suggests that com-plete financial networks are more stable than incomplete financial networks.It must be remembered, however, that the assumptions of the Allen–Galeresult are not satisfied by the Belgian banking system. Allen and Gale (2000)assumes that all banks are ex ante identical, whereas the Belgian banking sys-tem contains well capitalizedmoney center banks in addition to smaller banks.Also, the Allen–Gale result claims that greater completeness increased stability


ceteris paribus. The Belgian banking system by contrast experienced substan-tial changes between 1993 and 2002 in terms of the number, size, and balancesheets of the banks. Nonetheless, the estimated stability of the banking systemis a striking result.

DN make a number of other interesting observations about the structureof the Belgian banking system. They pay particular attention to the inter-national nature of the banking business and the fact that Belgian banks are wellintegrated in the international banking system. This leads them to distinguishanalytically between contagion having a source outside the Belgian bankingsystem from contagion originating in the failure of a Belgian bank. It turns outthat the extent of contagion caused by the failure of a foreign bank is somewhatlarger than that caused by the failure of a domestic bank. This result has to bequalified, however, because the most important foreign banks are very large,well capitalized and have very high credit ratings, so the probability of failureis correspondingly small.

Another interesting consequence of the integration of Belgian banks inthe international banking system is that several of these banks have very largeoperations outside of Belgium. As a result, their asset holdings are large rela-tive to the size of the Belgian economy. Our discussion of contagion so far hastaken no account of the safety net provided by governments and central banks.In the Belgian case, the very size of some of the banks might make it difficultto put together a rescue package for one of these large banks if it were to finditself in financial distress. Although UW estimated that the extent of contagionmight be large, at least if the loss ratio were large enough, the existence of asafety net might stop contagion in the early stages, before it reaches the criticalmass needed to spill over into a large part of the financial sector. By contrast,small countries with very large international banks may not have the resourcesto stop this process in the early rounds. As DN show, the risk of contagionappears to be small even in the absence of a safety net; but this may not be trueof other small countries, where some banks are literally “too large to save.”

10.8.3 Cifuentes, Ferrucci, and Shin (2005)

In our discussion of UW we pointed out that no account was taken of priceeffects, that is, asset prices are assumed to be unaffected by bank failures andthe process of contagion. DN and most other studies of this type make thesame assumption. We conjectured that if price effects were important, thedownward pressure exerted on asset prices by liquidations and/or hoardingof liquidity would reduce bank capital and accelerate the speed and extent ofcontagion. Cifuentes et al. (2005), henceforth CFS, have developed a model


of contagion in which price effects play an important role. In addition to theusual matrix of interbank claims, CFS add an asset market with a downwardsloping demand curve for assets. In this market, there are two channels forcontagion. The first is through the usual bilateral exposures in the interbankmarket; the second is through the effect of asset price changes on bank capital.Every bank is assumed to satisfy a capital adequacy requirement. When bankcapital is too low relative to the value of assets, the bank must sell some assetsin order to satisfy the capital adequacy constraint (it is not possible to raiseadditional capital, at least in the short run). The bank will first try to sell liquidassets, whose price is assumed to be fixed, but if it still cannot satisfy the capitaladequacy constraint it will have to sell the illiquid asset. The market for theilliquid asset has a downward sloping residual demand curve, so the more ofthe illiquid asset is sold by the banks, the lower the price. Contagion throughinterbank exposures works in the usual way. One bank failure creates a loss forthe creditor banks and reduces their capital. If the loss is big enough, capitalbecomes negative, assets are less than liabilities, the bank fails, and the lossesspill over to other previously unaffected banks. The new channel is different.When one bank fails, other banks suffer losses that reduce their capital. Ifthe capital adequacy constraint was binding, this would put the creditor bankin the position of having to sell assets to reduce the marked-to-market valueof its assets. At first, it may be possible to satisfy the capital asset constraintby selling liquid assets, but eventually it will be necessary to sell illiquid assets.If several banks do this, the asset price is reduced and this has an effect on banksin general. Other things being equal, a reduction in asset prices reduces theamount capital in each bank, possibly causing it to violate the capital adequacyconstraint. Those banks for whom the constraint is violated will be forced tosell assets themselves, thus increasing the downward pressure on asset prices.This all has a family resemblance to the story told in Chapter 5 and indeedit is very similar. The novelty of the CFS approach is that it combines theasset price channel with the interbank borrowing and lending channel to geta more powerful effect. The two channels run side by side, each reinforcingthe other. CFS do not calibrate their model to real world data, but they dosimulate the behavior of the model for reasonable parameter values and findthat the price effects greatly amplify the extent of contagion for appropriateparameter values. The analysis provides important insights into the factorsthat can increase the likelihood and extent of contagion and should provide aguide for future research.



There are a number of different types of contagion that have been suggestedin the literature. The first is contagion through interlinkages between banksand financial institutions. The second is contagion of currency crises. Thethird is contagion through financial markets. In addition to the surveys byMasson (1999) andUpper (2006) alreadymentioned,De Bandt andHartmann(2002), Karolyi (2003), and Pericoli and Sbracia (2003) contain surveys of thisliterature. Claessens and Forbes (2001) and Dungey and Tambakis (2005)contain a number of papers on various aspects of international contagion.Given the large number of recent surveys this section will be relatively brief.

Banks are linked in several ways including payments systems and inter-bank markets. These linkages can lead to a problem of contagion. We startby considering models of payment system contagion. Building on a locationalmodel of payment systems developed by McAndrews and Roberds (1995),Freixas and Parigi (1998) have considered contagion in net and gross paymentsystems. In a net payment system banks extend credit to each other within theday and at the end of the day settle their net position. This exposes banks tothe possibility of contagion if the failure of one institution triggers a chainreaction. In a gross system transactions are settled on a one-to-one basis withcentral bank money. There is no risk of contagion but banks have to holdlarge reserve balances. A net payment system is preferred when the probabilityof banks having low returns is small, the opportunity cost of holding centralbank money reserves is high, and the proportion of consumers that have toconsume at another location is high. Freixas, Parigi and Rochet (2000) usethis model to examine the conditions under which gridlock occurs. They showthat there can be gridlock when the depositors in one bank withdraw theirfunds, anticipating that other banks cannot meet their netting obligations if alltheir depositors have also withdrawn their funds. Rochet and Tirole (1996a)consider the role of the too-big-to-fail policy in preventing contagion. Furfine(2003) considers interbank payment flows in the US and concludes that therisk of contagion from this source is small.

As discussed above at length, Allen and Gale (2000) focus on a channel ofcontagion that arises from the overlapping claims that different regions orsectors of the banking system have on one another through interbank mar-kets. When one region suffers a banking crisis, the other regions suffer a lossbecause their claims on the troubled region fall in value. If this spillover effectis strong enough, it can cause a crisis in the adjacent regions. In extreme cases,the crisis passes from region to region and becomes a contagion. Eisenbergand Noe (2001) derive various results concerning the interconnectedness of


institutions. Aghion et al. (1999) also consider a model of contagion throughinterbank markets. In their model there are multiple equilibria. In one equi-librium there are self-confirming beliefs that a bank failure is an idiosyncraticevent and in the other there are self-fulfilling beliefs that a bank failure sig-nals a global shortage of liquidity. Lagunoff and Schreft (2001) study thespread of crises in a probabilistic model. Financial linkages are modeled byassuming that each project requires two participants and each participantrequires two projects. When the probability that one’s partner will with-draw becomes too large, all participants simultaneously withdraw and thisis interpreted as a financial crisis. Rochet and Tirole (1996b) use monitor-ing as a means of triggering correlated crises: if one bank fails, it is assumedthat other banks have not been properly monitored and a general collapseoccurs. Dasgupta (2004) uses a global games approach to show how a uniqueequilibrium with contagion can arise when banks hold cross deposits. Allenand Carletti (2006) show how contagion can occur through the market forcredit risk transfer. Van Rijckeghem and Weder (2000) document linkagesthrough banking centers empirically. Iyer and Peydró-Alcalde (2006) considera case study of interbank linkages resulting from a large bank failure due tofraud.

There is a growing literature on contagious currency crises and internationalcontagion. Masson (1999) provides a good overview of the basic issues. Hedistinguishes between “monsoonal” effects, spillovers and pure contagion.Monsoonal effects occur when there are major economic shifts in industrialcountries that impact emerging economies. Spillovers occur when there arelinks between regions. Pure contagion is when there is a change in expectationsthat is not related to fundamentals and is associated with multiple equilibria.Eichengreen et al. (1996) and Glick and Rose (1999) provide evidence thattrade linkages are important factors in the spread of many currency crises.Kaminsky et al. (2003) consider a long history of contagion across borders andconsider why contagion occurs in some cases but not in other similar situa-tions. Pick and Pesaran (2004) consider some of the econometric issues thatarise in distinguishing contagion from interdependence.

There are a number of papers that consider contagion through financialmarkets. King and Wadwhani (1990) consider a situation where informationis correlated between markets. Price changes in one market are perceivedto have implications for asset values in other markets. Calvo (2002) andYuan (2005) consider correlated liquidity shocks as a channel for contagion.When some investors need to obtain cash to, for example, meet a margin callthey may liquidate in a number of markets so the shock is spread. Kodresand Pritsker (2002) use a multi-asset rational expectations model to show


how macroeconomic risk factors and country-specific asymmetric informa-tion can combine to produce contagion. Kyle and Xiong (2001) present amodel of contagion in financial markets due to the existence of a wealth effect.Pavlova and Rigobon (2005) provide a theoretical model of contagion of stockmarket prices across countries arising from wealth transfers and portfolioconstraints.


Contagion is one of the most important topics in the area of financial crises.The idea that shocks can spread and cause a great deal more damage than theoriginal impact is one that is extremely important for policymakers. It is usedto justify much of the intervention and regulation that is observed. As we haveseen contagion takes many forms. Although there is a large literature on thistopic, much work in this area remains to be done. The same is true of all thetopics covered in this book!

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Index

Adams, John Quincy 3Agénor, P. 296, 297Aghion, P. 294Allen, F. 84, 90, 95,103, 124, 128, 147, 172,

176, 184, 193, 204, 212, 214, 218, 224,228, 231, 237, 238, 239, 246, 247, 250,252, 262, 280, 281, 290, 293, 294

Alonso, I. 95Argentina crisis of 2001–2002 17–18Arnott, R. 214Arrow securities 146, 150, 42Arrow–Debreu economy 41Asian crisis of 1997 1, 15, 260–261, 217–218Asset price bubbles

agency problems 237banking crises 247negative 236positive 236risk shifting 237without Central Bank intervention 254

Azariadis, C. 147

Bagehot,W. 3Bank capital

incentive function 193risk–sharing function 193

Bank of England 3Bank of Sweden 3Bank runs 74

empirical studies 96Banking Act of 1935 5Banking and efficiency 72–73Bannier, C. 93Barth, J. 213Benefits of financial crises 153Bernanke, B. 147Bernardo, A. 148Bertaut, C. 101Besanko, D. 192Bhattacharya, S. 94, 95, 192, 214Bill of exchange 126Blume, M. 101Bolton, P. 295Boot, A. 214Bordo, M. 2, 3, 9–12, 16

Bossons, J. 125Boyd, J. 18, 19, 228Brennan, M. 101Bretton Woods Period 1945–1971 10Bryant, J. 20, 58, 59, 74, 84, 94, 95, 96,

147, 149Buffers and bank runs 277Business cycle view of bank run

82, 95

Call loans 6Calomiris, C. 83, 94, 96, 147, 231Calvo, G. 260, 294Campbell, J. 100Capital regulation 191Capital structure

Modigliani–Miller theorem 203optimal 194with complete markets 201

Caprio Jr., G. 214Carey, M. 214Carletti, E. 294Carlsson, H. 90, 95Cash-in-the-market pricing 102,

110–114Cass, D. 147Chang, C. 233Chang, R. 230, 231Chari, V. 94, 95, 147Chatterjee, K. 97Chui, M. 229, 230Cifuentes, R. 291Claessens, S. 293Competitive equilibrium 159Complete markets 41, 70Cone, K. 94Constantinides, G. 98Constrained efficiency 182, 198Consumption and saving 27, 32Contagion

asymmetric information 260Belgian banking system

290–291currency crises 261, 294empirical studies 284–292

300 Index

Contagion (contd.)financial markets 294–295German banking system 284–290incomplete interbank markets 274overlapping claims 260payments systems 293price effects 291–292

Contingent commodities 40Corsetti, G. 231Costs of financial crises 18–19, 153Courchene, T. 215Crash of 1929 2Crash of 1987 100Credit and interest rate determination 243Crises and stock market crashes 5Crises in different eras 10Crockett, J. 125Currency crises and twin crises 229

Dasgupta, A. 294De Bandt, O. 293de Neufville Brothers 126, 147De Nicoló, G. 231Deflation 218Degryse, H. 290Desai, M. 258Dewatripont, M. 98, 213, 295Diamond, D. 20, 58, 59, 74, 94, 95, 96, 130,

147, 149, 228, 229, 230, 255, 262Diamond–Dybvig preferences 150, 116Dollarization 231–232Dollarization and incentives 226–228Dornbusch, R. 125Drees, B. 235Dungey, M. 293Dybvig, P. 20, 58, 59, 74, 94, 95, 96, 130, 147,

149, 230, 255, 262Dynamic trading strategies 156

Economywide crises 149Efficient allocation over time 27Efficient risk sharing 165Eichengreen, B. 9, 25, 294Eisenberg, L. 293Endogenous crises 148Englund, P. 14, 235Equilibrium bank runs 76Essential bank runs 85Excess volatility of stock prices 100Exchange Rate Mechanism crisis 229Extrinsic uncertainty 129, 148

Fama, E. 100Farmer, R. 147FDIC 190Ferrucci, G. 291Financial Crisis of 1763 126Financial fragility 126First Bank of the United States 3First Basel Accord 191Fisher, S. 125Fixed participation cost 101–102Flood, R. 229Forbes, K. 293Forward markets and dated commodities 31Fourçans, A. 16, 229Franck, R. 16, 229Frankel, J. 15, 235Freixas, X. 94, 213, 293Friedman, M. 58, 96Friend, I. 101, 125FSLIC 190Full participation equilibrium 120Fundamental equilibrium 129, 141, 148Furfine, C. 293Furlong, F. 192

Gai, P. 229, 230Gale, D. 84, 95, 103, 124, 128, 147, 172, 176,

184, 193, 195,199, 204, 212, 214, 218,224, 226, 228, 231, 232, 237, 238, 246,247, 250, 252, 262, 280, 281, 290, 293

Galindo, A. 231, 232Garber, P. 229Geanakoplos, J. 147, 198General equilibrium with incomplete markets

147Gennotte, G. 192Gertler, M. 147Glass–Steagall Act of 1933 5, 190Glick, R. 294Global games approach to currency crises

230Global games equilibrium uniqueness 90, 95Goenka, A. 147Gold Standard Era 1880–1913 10Goldstein, I. 90, 95Goodhart, C. 296Gorton, G. 4, 20, 83, 94, 95, 96,147, 231, 239,

247, 262Gottardi, P. 147Gramm–Leach–Bliley Act 190Great Depression 2, 190, 217

Index 301

Green, J. 57Greenwald, B. 214Guiso, L. 100, 101Gup, B. 215

Haliassos, M. 101, 125Hamilton, Alexander 3Hansen, L. 98Harris, M. 98Hart, O. 147Hartmann, P. 293Heiskanen, R. 14, 235Heller,W. 215Hellman, T. 192Hellwig, M. 95Herring, R. 213Hicks, J. 261Hoggarth, G. 18Honohan, P. 25, 233Hubbard, R. 151, 233

Illing, G. 296IMF 25Incentive compatibility and private

information 71Incentive compatible 131Incentive constraint 131Incentive efficiency 72, 175Incomplete contracts 154Incomplete markets 154Inefficiency of markets 66Inside money 216Insurance and risk pooling 48Interbank network

complete 263, 269incomplete 263, 271

Intrinsic uncertainty 129, 148Irwin, G. 212Iyer, I. 294Ize, I. 233

Jacklin, C. 94, 95Jagannathan, R. 95Japanese asset price bubble 15, 235Jappelli, T. 125Jensen, M. 241Jones, C. 6, 26Jonung, L. 259

Kahn, C. 94Kajii, A. 147

Kaminsky, G. 230, 231, 294Kanatas, G. 192Kanbur, R. 214Karolyi, G. 293Keeley, M. 192Kehoe, P. 147Keynes, J. M. 216Kim, D. 192Kindleberger, C. 2, 3, 20, 58, 126, 147,

235, 262King, M. 100, 294Kiyotaki, N. 147Klingebiel, D. 25Kodres, L. 260, 294Krooss, H. 4Krugman, P. 229Kwak, S. 25Kyle, A. 295

Laeven, L. 25Lagunoff, R. 148, 294Leape, J. 100Leiderman, L. 231, 232Leroy, S. 100Levine, R. 214Limited market participation 100–102,

114–124Liquidation pecking order 275Liquidity insurance 68Liquidity preference 53, 59–60Liquidity trading 100Long Term Capital Management (LTCM)

16, 127Lowenstein, R. 16

Magill, M. 147Mailath, G. 152Mankiw, N. 100Marion, N. 229Martinez–Peria, M. 25Mas–Collel, A. 57Mason, J. 96, 147Masson, P. 261, 293, 294McAndrews, J. 293Meckling,W. 241Mendoza, E. 260Merton, R. 100Mikitani, R. 15Miller, M. 296, 297Mishkin, F. 236Mitchell,W. 20, 58, 262Money and banking crises 228–229

302 Index

Money and risk sharing 218–223Moore, J. 147Morris, S. 90, 92, 93, 95, 230, 259Murdock, K. 215

Nalebuff, B. 214National Bank Acts of 1863 and 1864 4National Banking Era 4, 83Neave, E. 215Nguyen, G. 290Noe, T. 293

Obstfeld, M. 230Optimal currency crises 223–226Optimal monetary policy 256Optimal risk sharing through interbank

markets 266Outside money 216Overend & Gurney crisis 2, 3Özgür, O. 199

Panic-based runs 94Parigi, B. 293Parke,W. 100Participation and asset-price volatility

120–121Pauzner, A. 90, 96Pavlova, A. 295Pazarbasoglu, C. 235Pericoli, M. 293Pesaran, M. 294Pesenti, P. 231Peydró-Alcalde, J. 294Pick, A. 294Pigou, A. C. 216Polemarchakis, H. 198Porter, R. 100Portfolio choice 49Posen, A. 15Postlewaite, A. 95, 148, 259Prati, A. 93Prescott, E. 98Pritsker, M. 260, 294Production 36Pyle, D. 192

Quinzii, M. 147

Rajan, R. 94, 233, 229Real business cycle 147Regulation of liquidity 204

Reinhart, C. 230, 231, 234, 296Reis, R. 25Rigobon, R. 295Risk aversion 45

absolute 46relative 46

Risk pooling 55Roberds,W. 293Rochet, J. 90, 94, 95, 192, 293, 294Rogoff, K. 234Rose, A. 294, 296Roubini, N. 18, 231Russian Crisis 127Russian Crisis of 1998 16

Said, Y. 258Samuelson, L. 152Samuelson,W. 97Santomero, A. 192, 213Santos, J. 213Saporta, V. 25, 215Savastano, M. 234Sbracia, M. 93, 293Scandinavian crises 1, 14–15, 235–236Schnabel, I. 126, 147Schreft, S. 148, 294Schwartz, A. 58, 96Second Bank of the United States 3Second Basel Accord 191Separation Theorem 39Sequential service constraint 94Sequentially complete financial markets 170Setser, B. 18Shell, K. 147Shiller, R. 100Shin, H. 90, 92, 93, 95, 126, 147, 230, 291Smith, B. 25, 228, 233Sprague, O. 6, 96St-Amant, P. 215Starr, R. 215Starrett, D. 215Stiglitz, J. 215, 241Studenski, P. 4Stulz, R. 98, 214Sunspot equilibria 129, 140, 147, 148Sunspots and bank runs 76–77, 94Sylla, R. 6, 26

Takagi, S. 25, 26Tambakis, D. 293Tanaka, M. 215

Index 303

Thakor, A. 94, 214Tillman, P. 93Timberlake, R. 3Tirole, J. 213, 293, 294Tschoegl, A. 15, 235Turnovsky, S. 98Twin crises 9, 230–231

Uncertain bank runs and equilibrium 76–82Upper, C. 284, 293

Value of the market 63Van Damme, E. 90, 95Van Rijckeghem, C. 260, 294Vegh, C. 296Velasco, A. 230, 231Vihriälä, V. 14, 235Vines, D. 296, 297Vissing–Jorgensen, A. 101Vives, X. 90, 95, 226, 232Von Neumann-Morgenstern utility

function 44

Wadhwani, S. 294

Wallace, N. 94, 98Wealth and present values 20Weber, A. 296, 297Weder, B. 260, 294Weiss, A. 241Welch, I. 148West, K. 100Whinston, M. 57White, E. 242Wicker, E. 96Wilkins, C. 215Wilson, J. 5–8Winton, A. 94Worms, A. 284Wyplocz, C. 296

Xiong,W. 295

Yuan, K. 294

Zame,W. 186Zeldes, S. 100

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